Eck, take a look at rt 120 thru Wind River canyon going North out of Riverton, Wy. it's a beautiful ride and ends in Thermopilis, while in Thermopilis there are public hot springs you can swim in, also take care while riding thru town, the local deer population thinks it was paved for them.
Get on google earth, be sure places, photos and panaramio are checked, then get directions from Riverton, Wy to Cooke City, Mt. it's a great ride with lots to see. The East Yellowstone entrance is 5 miles from Cooke City.
You'll probably want to get a room in Cooke City unless you have time to get to lodging in Yellowstone. Getting caught riding the in dark in Yellowstone can be scary, those bison blend into the darkness and you don't see them till they are right there.
This route you'll see Wind river canyon and Chief Joseph Hwy. We rode each both ways just to see it from both directions.