The Texas State Attorney General has written an opinion regarding Texas Labor Code 52.061 as it applies to employees.
https://www.oag.state.tx.us/opinions/op ... ga0972.htm" onclick="window.open(this.href);return false;
I'm not aware of a similar opinion regarding non-employees. If you are considered an employee, then Texas Labor Code 52.061 authorizes you to legally store a weapon in your vehicle when parked on UTMB property. If you are not an employee, then you need to rely on Texas Penal Code 46.035(f)(3), which defines premises with regard to hospitals and educational institutions thus: "(3) "Premises" means a building or a portion of a building. The term does not include any public or private driveway, street, sidewalk or walkway, parking lot, parking garage, or other parking area."
So, regardless of whether or not you are an employee, you may legally carry your weapon until you arrive in the parking lot, and you may legally store it in your vehicle. I would seriously consider getting a safe to store it in. Cars can be broken into. You can find safes like these are Cabelas along with many other places -
http://www.meijer.com/s/gunvault-nanova ... kpid=NV100" onclick="window.open(this.href);return false;