I have a game I made for another jam that I want to update after that jam ends and I want to know if I'm allowed to post it here when I'm done. I figured this was low stakes enough that it shouldn't be a problem but I just want to double check to be safe.

I should note that the jam started early October so its within the date of this one.

Absolutely. Lots of people submit their games to multiple jams. As long as it's made in October and meets the other rules it's fine.