Platinum Sportswear in Georgia makes nice union-made shirts. They are not ringspun cotton though. They're more like a standard Gildan or Anvil shirt in cut and feel (not the Anvil model that Woot uses). Nice though, and quite inexpensive. I pay under $4 each shipped. Probably the cheapest domestic shirt you'll find.
A.S. Tees makes most of their shirt domestically. Most everything is ringspun cotton. They have a ton of women's styles which are cut like AA. I love their organic cotton shirts for men and women. A little pricey, but the cheapest domestic organic shirt I've found. It has a nice trim cut and super nice fabric. They don't have a lot of great domestic Men's options, but their Men's import shirt (7001) is awesome for the money. Super soft and slinky - nicer than AA, and cheaper too. Those are made in Haiti (usually).
Stay away from Bayside. The sleeves are really long and goofy. I can't wear them. They almost come down to the elbow. Maybe the women's are OK.
Royal's good, but not much different than AA in styles or price. They're kind of the East Coast "me too".
There's also Sun Apparel which has some sort of US/Mexico hybrid manufacturing. US fabric sewn in Mexico? I haven't used them.
Oh yeah, Anvil just started making a shirt in the US called the American Classic Tee. Imported fabric sewn in the US. Haven't used it. I know Anvil has a bad ap around here, but I think they are a good company with good products.
I haven't done any new research in 2 or 3 years. I'm sure there are more out there somewhere.