While I have not ridden the somewhat rough rail trail, supposedly it does go to West Yellowstone. Looking at the satellite view on Google, you can see the "trail" roughly parallels NF-478/South Fork Road, a gravel road. The trail has several soft sandy areas (due to ATVs) which make riding difficult on a mountain bike so a road bike would probably be extremely difficult to impossible to ride. In addition, the trail might have some bridges missing
forcing a turn around. If you do not mind gravel roads, then NF-478 would work. It eventually connects with the trail but again the trail might not be as easy as a gravel road. Note that this route is slower than the paved route.
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.