I think UT Austin may have more to offer at career fairs, but I don’t think having a degree from UA will hold you back.
Granted you may have to work a little harder if you have specific places you want to work, but the students we have known personally have done exceedingly well there. Three friends from Louisiana, all on full tuition scholarship with very different career goals. Fast forward 4 years. One is going to Harvard law in the Fall, the second has full scholarship + stipend to Tulane’s MD/PhD program, and my DS is going to put his MSF/BS Math to work as an Analyst at an investment bank. A good friend of his is a CS and he’s going to work at Google (where he interned last summer). Coincidentally, a friend from home went to UT Austin, also received his MSF, but has yet to find a job. (That’s not on the school, it’s just that he has very specific career goals and hasn’t been able to find position.) Point is, IMHO, it’s more about what you make of your time at school. If you are determined to work hard, earn the GPA and work to get experience, you will have good options at graduation.
You can contact the department and ask where their grads received offers or just use Google or Indeed and put company and college and see where alums are working. In terms of prestige, I’m not sure that really extends much beyond the first job, and if you get work experience before graduation that generally is the first consideration ahead of the university name on your diploma. I’m sure if you did some digging on Glassdoor and those types of sites you could get some feedback from an employer’s prospective. As far as the alumni network is concerned, I’m not sure how much that is a factor to be honest. But I know in my son’s case, the professors and advisors would often forward emails from alums about openings within their companies for internships/permanent placements. Likewise, social media plays a much larger role in recruitment these days and I think that helps to make more opportunities available.
The money is definitely an incentive, but the fact that you can really use your scholarship to the full extent is a tremendous perk. (Getting an override to take additional classes is quite common, esp for high stat students) Being able to take graduate classes under scholarship is an AMAZING benefit! And if you want to earn a dual major or MS degree through University Scholars you can make yourself even more attractive to potential employers. Plus meeting people from all across the country who are similarly motivated to do well at college is a great environment to encourage you to do your best 
Although we didn’t offer our son the difference in tuition, we did let him keep every penny he earned of his income from summer internships. So not only is he graduating debt free, he also starting his life with a sizeable nest egg! Roll Tide.