Not really one correct style - others may have different observations than me but most of the East Tennessee (or southern mountain in general) rifles that I have seen from that era - as well as even later - have tapered and flared "swamped" barrels. I have seen a few tapers and straight barrels as well, but most have been swamped.
Some actually do have pronounced taper and flare just a different profile than you usually see today - the narrow waist is often a little farther back, and they tend to still be muzzle heavy even with the swamp. Many of these rifle look light and slender due to the fine architecture but when you pick them up many are pretty heavy. Some are as wide or even slightly wider at the muzzle than at the breech, wihch is something you just don't see modern barrel makers doing.
Others are only very sligihtly swamped - you have to feel the taper and flare wtih your fngers or measure it as it is hard to see. Personally I have seen a lot of old guns that the owners assumed had straight barrels but actually turn out to be swamped if you feel or measure carefully.
Barrel lengths are (or rather were before they were cut as so many of these have been cut down) usually >43" - many are in the 45 to 47 inch range. An original barrel lenght under 42 inches would have been unusual for a mountain rifle based on surviving examples, of which there are many.