for what you are looking for, southern Missouri is excellent. Relatively low taxes compared to neighboring states for example.
OK it sounds like you are retired, have some income, and are looking to rent and hunt public.
In that case, it might be a good idea to concentrate on being close to Mark Twain National Forest, which is in 8 large blocks in southern Missouri.
The portion just SE of Springfield would’t be far from your timeshare, would give you easy access to a major city if necessary, and is just crazy nice country, look up the “Glade Top Trail” for example. If you like to fish it borders Bull Shoals like which was fantastic fishing at least a few years ago haven’t been down lately.
Good luck on your trip. Take a camera and your bow. You are coming down for prime fall colors, and the deer are starting to move around well.