Summer greens in the Deep South are a challenge...many recommended for summer use elsewhere (like chard, New Zealand spinach, lamb's quarters, dandelion, etc.) either won't grow at all through the summer or become impossibly fibrous and/or bitter. You need to consider truly tropical vegetables, with the exception of collards and possibly kale. Sweet potato greens are excellent cooked, and many people don't know you can eat them. My favorites from 20 years in Georgia are Indian or Malabar spinach (Basella), kangkong or water spinach (Ipomoea aquatica), and Surinam spinach or Florida lettuce (Talinum triangulare). They will all thrive on any amount of heat and humidity, provided they get enough moisture, and are useful both as salad and as cooked greens. Kangkong can be hard to find....try seed companies that specialize in Oriental vegetables. Talinum is as far as I know not available commercially anywhere....it's a West African weed now found throughout the Caribbean and Central America, and, sporadically, in central and southern Florida. The seeds of all three, like other really tropical species like eggplant and peppers, need a lot of warmth (80-90) to germinate well at all.....
Curly mallow is my personal favorite here. Never bitter. A favorite of Thomas Jefferson and George Washington. I think paying attention to ones climate is key on this topic.
Good King Henry, Salad burnet, tilia cordata, and black salsify are others. Eric Toensmeier's book "Perennial Vegetables" is a good one for this topic. I agree more shade and water in hotter climates.