That depends on your definition of quickly. Generally, plants that are annuals grow more quickly than perrenials. Plants such as Sunflower Helianthus annuus (Common sunflower) and Corn Zea mays can reach heights of over six feet in just a few months, but these are probably not what you had in mind. One of the fastest growing perrenials is said to be Bamboo whose optimal shoot growth can be 39" per day. Growing bamboo can be problematic, so you might want to consider something that grows more slowly.
An evergreen shrub that the "Green Gurus" frequently recommend is Wax Myrtle Morella cerifera (Wax myrtle). It is a 6-12 ft. shrub that can reach up to 20 ft., and is described by Floridata as being a fast grower. It can grow in full sun to partial shade, likes slightly acidic, moist, deep sands, loams, clay and is found all over east Texas.
To look for other possibilities, lets go to our Native Plant Database. After clicking on the link, scroll down to the Combination Search Box, and make the following selections: choose Texas under State, Shrub under Habit, and Perrenial under Duration. Check Sun under Light requirement ,and Moist under Soil moisture. Click the "Submit combination Search" button and you will get a list of 48 native species occurring in Texas that meet these parameters. Clicking on the name of each plant will bring up its NPIN database page that describes the plant's characteristics, its growth requirements as well as pictures.
Another option that would be faster growing would be an evergreen vine on a trellis, or on the fence if it is tall enough. Go back to the Native Plant Database, and proceed as before except choose Vine instead of Shrub under Habit. Click the "Submit combination Search" button and you will get a list of 24 vines from which to choose.
Three that might be possibilities: