Regarding TN, the Big South Fork area is a major equestrian recreation destination. There are a number of equine “subdivisions” in the Jamestown area. I get “come ons” from them regularly. I’m not a “subdivision” guy so I don’t even consider such places (and there’s no way on this Earth or any other that I would give control of a valuable asset to a “homeowners’ association”; those groups, as a rule, make the old Soviet Union look liberal).
Land in that area is not expensive. I know many people who own property that abuts the Federal land meaning they can access it very easily.
TN does not have an income tax on wages or pensions (but it does have small one on investment income). Sales tax is high, real estate taxes generally low. No personal property tax. Cost to register a one ton dually is $24/yr. (unless you require commercial plates or your county has a “wheel tax”). A permanant “semi” tag for a horse trailer is about $90 (for the life of the trailer).
Overall cost of living (food, energy, etc.) are also low.
Low taxes and costs means a low wages and a low level of public services. There’s no free lunch, don’t you know. 
The two most popular breeds in the state are the Tennessee Walking Horse and the Quarter Horse. Numbers are almost equal, with Walkers having a very slight edge.
There are a number of “trail riding clubs” around. Frankly, you have to be a bit careful 'cause while some are well run a couple I know of are “mounted drinking societies.”
Tamara can tell you more about the Big South Fork area 'cause she lives near it. 
Further east you’ve got the Smokies and a number of national forrests in VA, NC, and KY. Costs of living, land, taxes, etc. are significantly higher in those states.
To the south you have GA and AL. Both have active equestrian industries. Both have higher tax and cost structures (but AL is probably a bit less). Both are also much hotter in the summer time.
From what you write I presume that access to trails is very important to you. If that’s so then I would look at the access to public land, starting with looking at a map, identifying state/federal land, and researching equestrian access. Many places that look good at first glance have draconian restrictions on horses.
Good luck in your search.
G.