I hike in the Ohio-Kentucky-Tennessee area, and I have to think that a 3-season tent would work just fine. You might look at the REI-brand tents on their website; I just did, and found 8 or 10 one and two person tents on sale for under $200, plus some other brands. Moosejaw is offering 30% store credit right now, so a $300 tent would give you $90 of store credit to use for other stuff (you need to read how it works before you buy anything; it's not usable on all of their inventory, but there's a lot of good stuff that it applies to.) If you had $200 to spend on a tent, and $100 to spend on other gear, you might be able to get a more expensive tent and use store credit for the other gear.
Your post doesn't really tell us a lot about what you want the shelter to do. I used tarps a lot way back when, and they worked OK to keep precipitation off but let a lot of wind (and blowing snow) in, unless I really closed down the windward end - which can be a little tricky when you're trying to use hiking poles and get everything taut all by your self. I mostly used them in combination with a bivy bag, which served the same windbreak purpose a tent does. When I finally switched over to tents, I found I was much more comfortable in a tent in winter than I had been in a tarp, and for the same weight. A solo tent might be a good alternative to your tarp - it would probably be easier to rig when you're tired and cold, too.