Yellowstone September or October

Closures of hotels and also trails will have to do a lot with weather, as well as schedules (for the hotels). By late October, very little is open, though you could stay in one of the gateway communities (West Yellowstone, for example) and take day trips into the park, as long as the roads aren't closed due to snow. Some of the lodging in the gateway towns closes in October as well, but rates are better for those that are still open.

Fall color is already starting in the park, just FYI. And the elk rut has started. Temperatures in the park will be below freezing most nights within a few weeks.