The answer to that depends on what your idea of a Southern Mountain rifle is - now I am not trying to be a smart ass about this, so bear with me - to say Southern Mountain rifle is really a very generic term, kind of like saying "Kentucky Rifle". I think we get the idea of there having been a gun called the "Southern Mountain Rifle" from what we see available for sale as gun kits and semi custom rifles. The truth is that the guns built in the South were as varied as those built anywhere else
The gunsmithing families who made what we sometimes think of as the classic Tennessee rifles came to Tennessee from all over the place. The Beans were originally from Scotland and came to tennessee by way of Virginia. John Bull's family was from England, he was born in Maryland and had his apprenticeship there before moving to Tennessee. Many of them came from the Carolinas and Virginia. All brought with them what they had learned back home.
The type of gun I think you are talking about probably started to develop in the 1780's, and could have been built in parts of North and South Carolina, the south western part of Virginia and eastern Tennessee. Later as the families moved on this type of gun was being made in Kentucky, parts of Ohio - you get the idea
As far as the end of the time frame, it never really completely died out. Guns like these were still being used well after the percussion era, some even after cartridge guns came in, and people are still building and using them right now.
There were a bunch of great guns made in your home state, if you can find a copy of it check out James Whisker's "Gunsmiths of West Virginia". also Jerry Noble has four volumes of "Notes on Southern Long Rifles", lots of pictures of Southern guns and gunsmith listings