For a paved alternative, you might consider taking the ID-47/Mesa Falls Scenic Byway (all paved) south of Island Park, ID to Warm River to Ashton. Another alternative is to take Big Springs Loop/Fish Creek Road/NF-82 north and east of Island Park to Warm River but 17 miles of it are gravel. Obviously, you could stick on US-20 also.
While you did not ask for this, I will suggest to you anyway . From Yellowstone Park, you really should dip down to Grand Teton NP and Jackson, WY. The Tetons are fantastic! Be sure to take Teton Park Road via Jenny Lake to Moose. Then from Moose, take US-191 south to Sagebrush Rd. and then Spring Gulch Rd. into Jackson. Take obligatory photo of antlers then take WY-22/ID-33 to Victor, Driggs, Tetonia, and into Rexburg. Lots of great scenery.
Lastly, if you want to head south from West Thumb in Yellowstone, you can take the gravel Flagg Ranch-Ashton road to Ashton. It is the ACA Great Divide route.
Decisions, Decisions. Almost as much fun as actually riding the route.