Assuming you’re coming from Vermont,
Greyhound should be pretty easy for doing this. Your closest bus station to Gatlinburg is Knoxville, TN (also a fairly large airport there.) From there, you’ll need shuttling down to Gatlinburg one way or the other. There is a shuttle-bus service from Knoxville to Gatlinburg run by Sassafras Transportation, but you’ll need to call them for schedules since they don’t have a website; (865) 971-4075. You can also call one of the shuttlers on the ATC shuttle list to get not only to Gatlinburg but also to the trailhead (I’m assuming you’ll go in at Newfound Gap if you’re headed for Gatlinburg.) If you take Sassafras down to Gatlinburg, you’ll find it pretty easy in-season to get a hitch up to the trailhead.
At the other end, your closest bus station to Damascus is Bristol, TN (about 35 road miles from Damascus.) Shuttling from Damascus will be simple, as there are a host of people on the shuttle list there, as well as shuttling from the two outfitters in town. That will get you back on Greyhound and back home with no problems.
Hope this helps, and cudos for thinking to take public/mass transit first rather than drive.
Strategic