So I wanted to take some screenshots but the Steam Overlay wasn't working in a few games. I tried some methods I found on the net; FRAPS like programs interfering so closing them, disabling Firewall/Anti-virus, giving administrator priviledges to Steam,Overlay & game EXE files... None of them fixed the problem.

Then I tried manually running the overlay and managed to get it to work. Here's a method if you need it. (This requires manually doing it every time you run a game so I hope previous methods work for you instead)

You need to have a basic understanding of using command prompt on Windows. Open "cmd.exe" and navigate to Steam directory, "C:\Program Files (x86)\Steam" in my case.

Then run the game you want to activate the overlay on. We need its process ID (PID).

One way to find PID is the "tasklist" command on CMD:
> C:\Program Files (x86)\Steam>tasklist

This will give you list of running processes along with some extra info, like below:
svchost.exe 1956 Services 0 24.508 K
Steam.exe 3856 Console 2 92.280 K
steamwebhelper.exe 3444 Console 2 52.632 K
tru.exe 3040 Console 2 112.560 K

"tru.exe" at the end is the game I want to attach the overlay to and "3040" next to it is the PID number. This number is random everytime.

Take that number and run the following command with it;
> C:\Program Files (x86)\Steam>GameOverlayUI.exe -pid 3040 -manuallyclearframes 0

The zero at the end could be 1 or some other number, I don't know, didn't make much difference but try if you like.

Check the game, now you should be seeing the FPS counter and screenshot key should be working.

Have fun!

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TLDR;
You can add the steam overlay on a running game via command prompt.
Run the game and find its process ID (PID).
One way to find PID is "tasklist" command, gives running process names and their PIDs.
Using the game's PID, run the following program in steam folder from prompt;
"GameOverlayUI.exe -pid 3040 -manuallyclearframes 0"
3040 is an example PID, replace it with yours.
When you switch back to the game, the overlay should be working now.

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WARNING! As asked below, I can't guarantee if this gets you banned from VAC protected games or not. We are not running a cheat, it's Steam's own program so shouldn't be a problem but still, use at your own risk!