We were there in June. My favorite part was the Beartooth Highway. From the park go out the East Entrance. It will take you to Cooke City then on to BT highway. The road is steep in places but the views are breathtaking. You will see lots of snow regardless of time of year. If you continue you wind up in Red Cloud, MT.
also make sure you go through Lamar Valley just before sundown. Lots of wildlife. Wish I could go back.
Enjoy!