I see it as a state with in a state, but not something the US government wants to ever admit to. Because America is a democracy the level of Appalachia as a threat is shifting every so often. For you have very hands off governers, and government, then you have those who cannot stand the "Hillbilly Empire" selling illegal alcohol, guns, and even outright bypassing the government to sell their goods without tariffs to foreign markets, thanks in no small part to the strange fleet they have moving their goods. So on the one hand no one likes the trouble they cause: shooting any who come into their land, destroying roads, and track, even "collecting" taxes from areas they consider part of their realm. However on the other hand the black and gray market the hillbilly's have is far reaching, and supply everyone from the Governor of New York to the poorest factory worker everything from opium to lace curtains.
Think of it like Zapatista army in Chiapas. They are pretty much controlling these villages, and towns, and the government wants them out. At the same time they cannot go in guns blazing for political backlash, and due to the majority of the people supporting the so-called rebels.
So they would have this army maybe a dozen or so men for each encampment, but if pushed their numbers are in the thousands. While a large army could force them into a war, the hillbilly Empire doesn't go for that. Never ones for uniformity they go to war with everything from squirrel guns to smoothbore muskets. Yet knowing the land, and more importantly how to dismantle the supplies for American forces.