If you're legal to carry in the state you're in, you're legal to carry in the part of the SMNP that's in the state you're in.
One note: Don't go into any "government buildings"...IE naturalist office / ranger office etc. I'm not sure, but almost positive that "common & for the public buildings" like a stand-alone restroom is exempted. It muddies a bit when the bathrooms are attached to the nature center or what-not.
Enjoy the trip! It's beautiful out there! (watch out for bears...)
Also note: If you're in North Carolina, carry in their STATE parks is a no-no.
I'm not sure how they handle carry in National Forests (which are different than National Parks), State Forests, etc etc. I also believe that carry in restaurants / bars that serve alcohol is a no-no.
TN might also have some STATE park regulations and they were working on repealing the no-carry in restaurants serving alcohol. I think that's been repealed by now.