Someone on the website Quora just asked about visiting Yellowstone in October, so here’s what I came up with while surfing the National Park Service website for Yellowstone:
Facilities open in October are:
Lake Hotel and Cabins to Oct. 9
Mammoth Hotel and Cabins to Oct. 10
Old Faithful: Snow Lodge and Cabins to Oct. 16; Inn to Oct. 9; Lodge and Cabins to Oct. 5
All campgrounds close in September, except the one at Mammoth Hot Springs, which is open all year.
Almost all roads are open until November 7; Dunraven Pass and Beartooth Pass close on October 11.
Since the weather always turns cold and snowy more and more during October, visitors need to be prepared to dress warmly and could need snow tires in some areas. All thermal areas and most wildlife can be seen (bears are just beginning to think about hibernating), but visitor center hours are limited, some dining facilities are closed, and ranger programs have already ceased in September.
You forgot to mention that October can be absolutely gorgeous throughout the month (especially if the weather is nice)! However, wildlife changes its habits, at least compared to early June. The youngsters are no longer small babies, and many (birds and mammals) will have fledged by October. Bears may be farther from the roads and therefore less frequently observed as they move around frantically feeding in readiness for winter. However most of the herd and hoofed animals – bison, elk, deer, pronghorns, and moose – will still be around, and those that grow antlers will be a sight to behold.