If you look at the McCombs thread there may be other folks who are waitlisted and they may tell you how it works. Some schools only let you wait until the next deposit deadline,

When you are waitlisted, it is better to assume that you will ultimately get rejected by the school than to assume that you will ultimately receive an admission offer. I know that the school you get WL'ed to may be your first choice, and that all the admissions consultant blogs try to say that you still have a chance to get in, but at most schools, the acceptance rate from the WL is lower than the overall acceptance rate, so I rest my case there. In other words, go all out for your top choice, but don't be super disappointed if you are dinged later on.

After all, the school likes you, but doesn't like you enough. You also applied in the last round, and there are fewer spots available for the taking, so your chances may be lower. Unless your previous GMAT's weren't anywhere close to the 10th percentile range, if you know that Texas allows waitlisters to re-take the GMAT, then it may have been worth applying earlier so you can get an earlier decision (even a WL), and then improve your app after the WL if that's what you get.

Most schools give about a month to admitted applicants to submit their deposits (note that I didn't apply to Texas). If McCombs allows you to submit more info, do so if it will help your candidacy. It won't hurt you if you don't send anything, but it will help you if you are able to improve your GMAT for example or if something new happened at work like a promotion or a new project development (though the latter doesn't just happen to everyone).