So after an hour, I have a partially-completed game: the first part is playable but it doesn't have an end. I am planning to finish it, but should I post the completed game, the version I had done after one hour, or both?
That's already great that you have done something so far! The jam intention is about having simple ideas and implementing them quickly. The one hour deadline is there to put a scope on the ideas. However, it's up to you if you strictly want to reach it. Personally I've exceeded it... so I'd suggest, as long as it's about finishing to implement your ideas, and not adding any new features, not going to a vicious circle of always polishing the game... adding some more time is okay.
Please remember that there is a theme for each week, and that you should have created your game with the intention of somehow relate to the theme.