Open-Jaw Flight Ticket Prices

$3900 for six people seems pretty good to me, even for winter. I don't think there is ever any trick to timing a purchase, just when you see a price you can live with and a schedule that seems workable.
If you have the option, then I would not fly over night, but many flights are timed that way.

After searching these forums, it looks like I missed the very best window for purchasing flights (the 330 days before the return flight).

There is no very best window. Flight prices go up and down according to demand. $3900 seems like a good price for 6 tickets over the Christmas holidays; that is only $650 a ticket and I have not been able to get a ticket at that price since before the pandemic.

Check the airline’s change policy. I bought a ticket on United in February. In March, the price dropped by $190. I was able to get a United credit for the difference which I can use on another United flight. If your airline has such a change policy, you may be able to get a credit if the price drops.

I prefer to arrive in Europe as late as possible to minimize the amount of time I have to stay awake the first day.

I have not seen any Fall Winter airfare sale solicitations from the carriers in my email.
But $3900 for 6 is excellent. I hope you are not looking at prices from a third party (cheaptickets.com) though.

What time of day do you prefer to arrive when traveling from U.S. to Europe, to make the jet lag as bearable as possible, especially with kids?

There are only daytime flights to London arriving closer to bed time; those are the best.
Any flights leaving later in the day getting you to Europe closer to noon and hotel checkin hours.

I booked far in advance on American Airlines website for a direct flight to Paris. About three months out I checked prices and they had dropped. I was able to go online with American through their customer service chat and requested and was given a $120 flight credit that I will use this summer for a family vacation to California. It took about 5-10 minutes. So might be worth it to book direct with an airline you have a good chance of using again within a year if the same thing happens to you.

Am I likely to get anything cheaper than that? We're still pretty far
out, but I don't know if that makes a difference these days.

Maybe, but that's the gamble you take booking now or waiting. There is no best window. You can use a price tracking tool like Google Flights and see how fares change, but if you're comfortable with the price you have then I would book it immediately before it goes up (and IMHO, $650 round trip per person during the holidays is a very good bargain - I've never paid a fare that low to Europe any time of year).

As Laura mentioned, check the policy of the airline, you may be able to get a credit if the fares go down. I did recently learn doing that very thing with Delta that the credit attaches to each person individually, NOT the person who paid for the tickets. So read the details very carefully.

The price was so good that I looked at our dates and destinations.

I didnt spend a lot of time but it appeared that the $640 tickets all had a problem on the return flights; generally long layovers. To avoid that the next best was $850 to $860 a ticket.

I stopped at that point, but I am certain that the prices shown are what ever United and Lufthansa call their basic economy, and thats fine, but i dont think that will include a checked bag or the ability to pick seats before checking in without paying another $100+ on each ticket for seats. If you upgrade to the next level to get the checked bag and to be able to choose a seat (might still cost) I suspect that you will be around $1200 a ticket. Someone else mentioned changing for a credit when the prices went down. Well that person purchased an even higher grade of ticket and for that you will be talking even more.

EDIT: Okay, I played with it. My prediction is $1100 per ticket plus the possibility of having to pay for seats, but you do get a checked bag.

Agree there is no "best time", too many variables. I had been watching ATL - Paris for this summer for months. The fares were (Main Cabin) nearly $1800 a person. Suddenly in mid March the fares decreased to less than $800 for Main Cabin with adequate seats. Likely due to softness in booking as muchly discussed in the news lately. We are flying in two weeks Mid May.
I don't think I have ever seen a best fare 330 days out other than frequent flyer seats. Airlines change fares daily for all sorts of reasons. Google Flights reminders and Going.com are good sites.
Searching google flights right now I see multiple options at roughly $950 for Main Cabin at this moment. Fares for basic economy are in the sub $600 range.

As Mr. E suggested, your prices might be basic economy.
Be sure to KNOW what you are buying before you make your purchase.

Basic Economy is priced lower because it comes with a number of restrictions, including seating (often seated last and cannot choose your seats in advance, so you might not be seated together) and sometimes on baggage, although rules might be different to Basic Economy on an international flight.

Those are surprises that you don't want when you arrive at your departure airport.

Yes, those prices are definitely basic economy. We're traveling on a tight budget, so I understand that we won't get to choose our seats or have a free checked bag. It seems the drawback is a really long layover (12-14 hours) at an airport on the way back (Madrid is the one I see). How difficult is it to leave the airport, go sleep for 8-9 hours at a hotel, and get back to the airport for the rest of the trip home?

Yes, those prices are definitely basic economy. We're traveling on a
tight budget, so I understand that we won't get to choose our seats or
have a free checked bag.

I cannot imagine flying with 4 young people and not choosing a fare that allows you select seats so you are sitting near each other. Are you comfortable with your 10 year old (or any of your kids, for that matter) sitting alone in a middle seat between strangers for 8-9 hours? Plus managing their own carry on bag because they could be quite a distance away from where you're sitting could even board at a different time than you? Unfortunately you can't count on the kindness of strangers to be willing to change seats with you so you can sit together. That's one of several possibilities that would make basic economy flight for your family situation a little rough.

It seems the drawback is a really long layover (12-14 hours) at an airport on the way back (Madrid is the one
I see). How difficult is it to leave the airport, go sleep for 8-9
hours at a hotel, and get back to the airport for the rest of the trip
home?

I understand budget concerns, I live on a fixed income. But the amount of money you might "save" by choosing basic economy would be sucked up by paying baggage fees, plus if you leave the airport on a long layover you'll probably need more than one hotel room (you will be hard pressed to find a room for 6 anywhere), plus transport to/from airport (probably need a private transfer unless you want to try public transit). Not to mention giving up an entire extra day to traveling home at the end of a pretty long trip.

Maybe pause a moment and track airfares for a few weeks. Do price tracking for the best option for main cabin (regardless of cost), as well as the not so great ones with basic economy. Also you said that your dates are "around" those listed, so also look at different days of the week. Changing plus or minus a day can often save hundreds (though you're traveling at a peak time so you may not get a deal).

Nope, on Lufthansa "Basic Economy" is a step up from those prices. Those prices are "Economy Light" LOL

You are willing to fly the way I do when ever possible. It's a few hours on a plane and you end up someplace marvelous.

I believe it does allow you to choose seats at checkin. But not 100% on that with Lufthansa. Oh. I also bet Lufthansa tickets and Lufthansa equipment is a better choice than the United options.

As for a layover hotel, both Munich and Frankfurt have hotels within a few hundred meters of the terminal buiding. But when you pay for the hotel, the more expensive flight is cheaper or essentially the same price.

I doubt the tickets were cheaper 2 months ago. These are very good prices. I would buy them soon. Directly from Lufthansa.

I cannot imagine flying with 4 young people and not choosing a fare that allows you select seats so you are sitting near each other. Are you comfortable with your 10 year old (or any of your kids, for that matter) sitting alone in a middle seat between strangers for 8-9 hours?

I don't know what airline you are flying, but here is some information on the US carriers regarding their commitment to family seating regardless of ticket type. Some agree to it and some don't.

https://www.transportation.gov/resources/individuals/aviation-consumer-protection/airline-family-seating-commitments-dashboard#conditions

And Lufthansa policy regarding families being seated together.

https://www.lufthansa.com/ca/en/travelling-with-children

And here's the link to sleeping at airports for Madrid. Might be some useful information in there.
https://www.sleepinginairports.net/guides/madrid-barajas-airport-guide.htm

As for jet lag, we have arrived with our kids late morning, been able to check into an apartment, and nap for 1 to 2 hours followed by a shower and going out to walk somewhere late afternoon and then go to bed early the first night. You can also try to shift your schedule by a few hours before you go.

First, that price for 6 people is probably about the best you will find. I do very occasionally see round trip tickets for $550 on sale but not open jaw.

Second, the low prices come with trade offs and only you can decide what’s worth it to you. The cheap ones won’t be cancellable - but you will probably want some form of trip insurance anyway that will cover a loss due to cancelling for health reasons.

Price versus arrival time? You probably won’t have a lot of choice and you can make it work.

You’ve gotten some good food for thought here. I will toss out a few other thoughts.

You could consider a combo of ticket types. Maybe 2 at the higher price (like Mr. É gave you) to allow for 2 combined checked bags and two seat reservations for a parent and the youngest together. That just means you will have 2 locator numbers.

Lufthansa has a pretty strict carry on size and weight policy that you would need to know and abide by. It IS possible to add a checked bag (at additional cost) on the way home, should you need that.

For long layovers: I have actually started doing this occasionally by choice. Sometimes I don’t like my connections without an overnight layover - or I don’t want to get up and be at the airport at dark:thirty. It does add cost that would need to be figured in to your overall budget.

I suspect you could get by with 2 hotel rooms in Madrid. I was just looking for myself last night and came across this website if it helps. https://www.esmadrid.com/en/hotels-close-airport
If you go this route, 1) try for the longest overnight layover possible. Sometimes I will choose a 22 hr one as opposed to a 12 hr one - to not be rushed at my layover and to get a good night’s sleep, not just a few hours. 2) make sure the hotel has a free shuttle that meets your needs. 3) Madrid had some affordable hotel options nearby but it would probably eat up the ticket savings from one person’s cheap ticket.

The 330 day out window is what you need for using miles or points but these days isn’t particularly relevant to using dollars.

Good luck!

This is all giving me so much to think about and consider. Thank you so much!