From Natchez Trace, go up to Land Between the Lakes wander back by Fort Campbell and stay North of 40 from there to Crossville by Catoosa Wildlife Management Area out to Big South Fork. Take a look at Cumberland Gap then over to Blue Ridge Parkway down to Smoky Mtn National Park. NOTE: there isn't an e in Smoky.
I think it's a mistake to take very many backroads in Middle or West Tennessee. There's just not that much to see that is not viewable from 40. Unlike many states, there really aren't continuous towns along most of 40 in TN. NOTE: I grew up in East TN and traveled all over for the 30 years before I moved West.
I'd also consider touring a dam (Army Corp or TVA) while you are in TN. They have visitor centers and tours inside. A good thing to do on a very hot day.
