I put together a route a few years ago borrowing liberally from the RUSA regional brevets from near the Trace over to Robbinsville, NC. Leipers Fork school is about a half mile from the Natchez Trace, and the route skirts metropolitan Nashville. Robbinsville is some 40 miles from Cherokee, the southern end of the Blue Ridge Parkway. If you ride the tail of the dragon, you can stay on NC 28 and skip Robbinsville, but you might like some services about then. You'll still have a few miles on the 19/74 racetrack, but the shoulders are pretty wide through there. Either way into Robbinsville, from Tallassee or Tellico Plains, take lots of water -- there's one bar on the northern route, a campground near the bottom of the southern, and plenty of climbing.
https://ridewithgps.com/routes/30047819Disclaimer: I've cycled maybe 40-50% of that route, and driven perhaps 25% more. I have no qualms recommending what I've biked. The Pikeville-Daytona stretch was still being widened when I last crossed Walden's Ridge there, so I don't know what it's like now.