Ask Mr. Smarty Plants

Here is a recent Mr. Smarty Plants answer to a question similar to yours. In brief, this answer recommends several native sedges (genus Carex) and two other native plants. Since you are interested in fast growing and sedges are likely to be a bit slow, I will focus on the other two.

The first is a personal favorite of this Mr. Smarty Plants, Calyptocarpus vialis (Straggler daisy) aka horseherb, lawnflower, or creeping Cinderella (in Australia). This plant will do best in shade or partial shade, but does well in sun. In addition to fast growth, it has a couple of things going for it: it is attractive, with bright green foliage and tiny, yellow, daisy-like flowers, and it is readily available. Unless your yard is very well manicured, you probably already have patches of it growing somewhere. You don't have to buy it, just dig some up, divide it and replant where you want it.

The second plant is Phyla nodiflora (Texas frogfruit). This plant is very tolerant of sun and is fast growing. It produces tiny white flowers. Like straggler daisy, you can probably find some growing wild in your area.

Both of these palnts are native to central Texas and well adapted to our heat, drought and occasional floods. Both will go dormant in most winters, but come right back when things warm up.