Austin is the capital of Texas and is also a center for film and home to the University of Texas and Formula 1's Circuit of the Americas raceway.
Here I am providing you name list with address of the clinics in Austin but these are open or not for 24 hours a day I am not sure:
Texas MedClinic  
12319 N Mopac Expy, Austin, TX 78758, United States
Phone:+1 512-835-5577

Austin Emergency Center  
3563 Far West Blvd Suite 110, Austin, TX 78731, United States
Phone:+1 512-481-2321

Texas MedClinic  
9900 S IH 35 Frontage Rd, Austin, TX 78748, United States
Phone:+1 512-291-5577

First Choice Emergency Room  
10407 Jollyville Rd, Austin, TX 78759, United States
Phone:+1 512-628-0470

Austin Regional Clinic:Far West  
6835 Austin Center Blvd, Austin, TX 78731, United States
Phone:+1 512-346-6611

MedSpring Urgent Care  
3906 N Lamar Blvd, Austin, TX 78756, United States
Phone:+1 512-861-8040

Neighbors Emergency Center - Lakeline  
12701 Ranch Rd 620 N, Austin, TX 78750, United States
Phone:+1 512-258-1195

NextCare Urgent Care  
6001 W William Cannon Dr #302, Austin, TX 78749, United States
Phone:+1 512-288-3627

First Choice Emergency Room  
15100 FM1825, Pflugerville, TX 78660, United States
Phone:+1 512-600-9888

Austin Emergency Center  
4015 S Lamar Blvd, Austin, TX 78704, United States
Phone:+1 512-774-5780