Is the Keeper from GG allowed to leave his starting area and then come back later to use his Quantum-Cosmic Jump trait? Or can he only use this trait if he has yet to leave the starting area?

Very helpful if you quote the power you're asking about so people can see the text:
You can give him a power action to use Cosmic Quantum Jump if he occupies your starting area. That's the only check: is he currently in your starting area? If the answer is yes he can use CQJ.

Well let us look at how the power is stated:

It says once per game, so it should not matter at what point in the game he does the jump, but he can only do it once.