Layover at IST

2 hours to town and back. Going back through security another hour. Leaves you 3 hours, best case. Check into the Yotel Hotel on the gate side. Get some good rest, and you will be fresh for wherever it is you are going.

With 6-ish hours, you don't really have enough time to comfortably get from the airport into the city, do anything meaningful, and get back out to the airport (clearing at least 3 security checks). If you knew your way around, and did everything right, and got lucky, you might be able to pull off a quick "drive by" peek. I know that sounds crazy, but the airport is absolutely gigantic (it'll take you some time to get off your flight, through the formalities, and out of it) and it's a long, long way from the city (at least an hour drive each way under the best of conditions in a private car or taxi), and traffic in/around Istanbul is usually terrible, and is often downright nightmarish.

With 6 hours, I would not risk it. Find a comfy spot in the airport. The Turkish Airlines lounge is spec-fricking-tacular (there are decent places to nap there if you can score one); we always look forward to spending time in the TK lounge (my spouse demands at least 3 hours there on any layover or else I'll hear about it). You may have access to one of their lounges (depends on your ticket & status). In your case, it's worth checking to see if you have/can get access. The airport is vast and there's no shortage of airporty things to explore.

One option. If you can secure a guide with a car to pick you up and return you and if you have one or two certain things to see and do. Won't be cheap. What time do you arrive in IST? That will have a lot to do with it.

Let's see, if you wanted to take any amount of time to leave the airport, that's two security checks (one at the airport entrance, and one near immigration), and then two passport check queues (one for entering Türkiye, one for leaving it). And then there's the fact that it's a massive terminal, typically with long walks between gates and everywhere else.

Turkish Airlines business class I believe has its own immigration/security queues.

Getting to/from Eminönü/Cağaloğlu (the prime hub of tourist destinations in Istanbul) takes at least 40 minutes one-way. And don't forget about your flight's boarding time, which reduces the layover length once again.

You won't have time to make it into Istanbul and back.

There is a lovely museum inside the airport that pulls from other museums throughout the country.

The one thing you have going for you is IST is large but incredibly efficient. I've done 60 min layovers there and made my connection. Not fun, but did it. But I love the hotel and the food courts and bars are excellent.

Kilyos is a seaside tourist town worth a visit! It's about a half hour drive from IST and has several restaurants, souvenir shops and a good beach. We spent 2 nights there back in early November and my wife said it was her favorite part of our 10-night trip. IST is big but when we were there not very crowded and efficient to get through. With a 6-hour layover you would probably only have 2-2.5 hours at best in Kilyos unless you cut it close.

Don’t forget that, if you are a US citizen and leave the airport, you will need a visa.

Which is a few dollars done online. But, yes.

Fifty dollars, actually.

It seems to have slipped under the radar, and hasn't been publicized much but Turkey no longer requires a visa for American tourists staying under 90 days. https://travel.state.gov/content/travel/en/international-travel/International-Travel-Country-Information-Pages/Turkey.html

Not mentioned so far is that the footprint of the airport is so large that there may be a very long taxi distance to the gate. I believe published arrival times are for touchdown, not being parked at the gate, so you may not have as much time as you think. My flight from Warsaw seemed to be heading to the gate by way of the Greek border.

Having experienced Istanbul traffic, there's no way I'd leave the airport with such a short layover. I wonder whether this schedule change puts you in a position to get a free ticket change. Turkish Air has two flights a day (or did in October) to Washington-Dulles; I don't know what the situation is to other US destinations. Perhaps you could switch to a later flight out that allows you enough time to get out of the airport.