Ah, got ya. Yes, you will have a spot in the garage.
Explaining it as best I know- UT has a certain number of spots and the remainder are public. The number of public spots vs. cars in the garage is electronically tracked so the attendants are aware of how many spots are left at any time (its on a sign at the entrance). In the absolute worst case scenario that a huge number of the non-UT spots were taken, the remainder *should* still be open for UT passholders only, even if the garage was "sold out".
UT has marked spots in the garage for the Speech/Audiology center, and those are almost never taken because of the signs on them.
Granted, I said *should* because humans are involved and it appears UT hasn't exactly bathed themselves in glory this summer as far as assignments go. But I think you'll be ok.