Teton area rides
If you take the scenic loop from Moose, don't slab it to Moose - go to Teton Village (maybe lunch at the Mangy Moose - recommended) and continue north on gravel/old asphalt to Moose. Check out the visitor center there. If you're a wine/microbrew fan, visit Dornan's store at Moose (just east of the Snake river - the scenic loop goes west).
For a longer ride, try the Rockefeller Parkway between Teton and Yellowstone parks - north through Teton park, then take a left at Flagg Ranch (another nice restaurant) onto the Ashton-Flagg Ranch road (AKA Grassy Lake Road, AKA Reclamation Road) for ~30 miles of gravel/rocks/dirt and scenery - suitable for a large GS. Right on Idaho 47 along the Warm River - check out the Mesa Falls overlooks. If you have all day, continue to West Yellowstone and loop through YNP to return, otherwise backtrack.
Beartooth Highway? Long trip! If you take it, Go through YNP, out East entrance to Cody and north on the Chief Joseph scenic byway to the Beartooth. Museum in Cody is worth a stop - western art/Buffalo Bill memorabilia/old gun collection.
Nice scenery and twisties south of Jackson along the Snake river - left at Hoback Junction and continue south along the Hoback River.
So many choices, so little time.....