I'm not in the South, so I can't give you first hand recommendations, but I buy my garlic from The Garlic Store, and their website has tons of info on the different varieties.

Here is a link that might be useful: The Garlic Store

I've never grown them but some of the creole garlics might be what you're looking for. I've tasted Creole Red and it's pretty good - started off mild but the taste really built.

In all respect, Beth, let me suggest that you first peruse the backpages. Your question has been addressed and answered numerous times, here, and you might same a lot of time by glancing at previous threads.

Along the way you'll also pick up growing and storing tips, and other useful data.