I'm going to let others recommend their favorite groundcovers, but I would like to suggest that you avoid invasive (environmentally damaging) plants. Many mints fit into that category.

There are many types of mints. Some are native, some not. Many that don't pose a danger, some that do. Mentha x piperita (peppermint), for example, is listed as a "Significant Threat" to native flora and fauna by the Tennessee Exotic Pest Plant Council (TN-EPPC). Mentha spicata (spearmint) is listed as a "Lesser Threat", but should probably still be avoided. Mentha arvensis is an example of a native Mentha species that might be a good garden substitute for some of the more harmful species. All of the Pycnanthemum (mountain mint) genus are native to North America. There are also other genera that could be considered. It's important to remember that many non-native species are just fine. Mentha suaveolens (apple mint), for instance, is not native to North America but does not show any signs of being invasive here in Tennessee. It can be aggressive, but aggressive is not the same as invasive. Plants that are aggressive but not invasive tend to make the best groundcovers.

Here is a link that might be useful: TN-EPPC's list of Invasive Exotic Pest Plants of Tennessee

I love my creeping thyme, irish moss, and variegated ivy. Periwinkle should be avoided, it gets crazy. :) purple clover is nice, but sometimes dies out in heat. Ask someone from your local garden center, they would know as well. The lowes on hwy 64 in Bartlett has a very knowledgeable associate if you are in memphis.