Private guide for Hermitage

We went into St. Petersburg on a cruise ship and spent two days with one of the 50 or more fantastic small tour agencies. Our bus only had 10 people on it, and our tour guide had a graduate degree in Russian history. With such a small van/bus, our driver took us down back alleys and small streets in order to beat all those 45 passenger full size buses to the Hermitage and all other popular tourist sights. The guides are independents and may work for two or more tour agencies.
You can find names of the tour agencies online and on TripAdvisor. I'm sure they offer group and private tours too.

Check with Anastasia Travel. I am sure if they are able to customize group tours, they can offer you a guide for the day.

Finding a guide is not going to be a problem,so I'll address the crowds/after hours issue:

On Wednesdays and Fridays the Hermitage closes at 9pm. Most cruise ship groups are out by six, so the only truly crowded area will be the Peacock clock room (the weekly winding of the clock is at 8pm every Wednesday) on the second floor and the enfilades leading to it. Also, they would typically have some kind of performance in the theater, so the theater hall will be blocked off, and you will need to find a detour.

So, if I wanted to experience the Hermitage with tolerable crowds, I'd go on a Wednesday, start at about 3 pm with the impressionists and post-impressionists at the General Staff building and relocate to the main building around 5:30. Skip the Peacock!
Here's the map: https://www.hermitagemuseum.org/wps/portal/hermitage/trip-planner-map-view/
You can also search for individual pieces you're interested in and plan your route accordingly (but I do have to say I still get lost there almost every time I go :-)

Thank you all very much for the helpful advice. I may divide my Hermitage visit into two half days, and I like the idea of visiting the Impressionists in the other building first. I can't remember where I read about someone who was able to enter the Museum (with a guide) before it opened in the morning. Has anyone done this, or know someone who has?

Exeter mentions private after hours tours. Scroll to “After Hours Tours of the Hermitage” and 1-800 number. Exeter International is in Florida, scroll all the way down to see their full contact and email. https://www.exeterinternational.com/blog/bloghermitage-museum/

I'd guess most companies dealing with cruise tours will offer an early access tour. The options are listed on the official site, and some tours are truly "special" (such as visiting the Diamond Room, attending a lecture/seminar by restoration artists, a classical music concert, etc), and well worth the money.
That said, the vast majority of "early access" tours is just a way for the museum to accomodate the growing cruise crowds: it's not a "free roam" experience; the tours are limited to certain rooms only, and they are anything but "private".
PS. Here's more quotation marks for the gods of quotation marks: "parenthesis", "parenthesis", "parenthetical insertion".

Thank you again! I checked the Exeter website and it looks promising. I forgot to mention I am not on a cruise, but traveling independently, with my son, (we will very soon send in our Visa applications).

Sounds like a good starting point.

Once you get a quote from Exeter, it would be interesting to compare it to the ones from Bestguides-spb, alla tour, red October or any other better-known local tour company.

As a point of reference, a regular Wednesday DIY tour would set you back about 2000 rubles, or $30 (₽800 entry fee, ₽500 audioguide, and ₽350 a separate ticket to Gold/Diamond Storage room, open 11am to 3 or 4pm). I have to say, though, that logistically it may prove somewhat challenging if the goal is to avoid standing in line - the tickets to "storage rooms" are not available at the ticket machines, the storage rooms are in different wings of the building, and once you leave a building, you can't go back on the same ticket.