Welcome to one of the joys of the south!
The best thing I have personally found for my birds is to plant rosemary bushes where they can get to them. You may need to protect the roots to keep the birds from digging them up however. I cut a piece of hardware cloth a foot around with a hole for the plant to come through, put it an inch or two below the surface and cover with dirt when you plant, keeps the birds from digging them up. You can also put large rocks or bricks on the cloth to help keep it in place. The oils and scent from the plants seem to discourage the gnats and my birds spend a lot of time in and under them during gnat season. They are fairly care free plants as well, once established. You have to be careful of spraying anything on your birds on a regular basis as many things can cause them respiratory irritation, particularly if they are aromatic. Besides being horribly time consuming.