There are 2 native cultivars of sunflower in Texas, I have only planted the Maximilian and planted in early Spring. The other cultivar is called the "Annual Sunflower" I had a friend that would plat that in December.There area few hybrids too.
From A&M, great article toohttps://tpwd.texas.gov/publications/pwdpubs/media/pwd_rp_w7000_1152.pdfGROWING MAXIMILIAN SUNFLOWERS
Aztec Maximilian Sunflower
is a variety released from the USDA Natural Resources Conservation Service’s Knox City Plant Materials Center in 1978. It
may be planted in range seeding mixtures during the spring at approximately ¼ to ½ pound per acre to a depth of 3/8 to ½ inches.GROWING NATIVE ANNUAL SUNFLOWERS
Initial plantings of native annual sunflowers should be conducted during the fall or early winter in a well-prepared seedbed for growth during the following spring and summer.