If you search the Internet for "Social Security Number Trace" you can learn more about this process. Basically various data collection agencies, based from records like credit/job applications, etc. where your name, address, and SSN are mentioned together, enable with a person's SSN to find other addresses or names that have been reported and associated with the number so they can use that information to more thoroughly search your background. It is
not an actual search of the Social Security Administration's database (only authorized agencies can do that).
If your SSN is the same as before (which unless you went through an arduous process to get it changed is the case), then such a trace may very well show your prior name depending on what age your name was changed* and how long ago it was.
*A former name that was changed before you'd have any work/credit/other "adult" records probably won't show up (and if it does that's a sign of identity theft from when you were a child); same thing if you immigrated to the U.S. and your name was changed beforehand. Speaking of which, if they do see an opposite-gender name that was used, they may well assume that it wasn't you being TS but rather your information was mixed up with that of a relative or other person (these traces are far from being error-free - a mistake in the aforementioned name/address/SSN combination on an applicable form or a data entry error can result in other people's information being crossed with yours or vice-versa, as well as when someone else is trying to illegally use parts of your identity).
To get an idea of what I'm talking about, you can search (don't pay for anything, just use whatever searching you can do for free) at sites like
intelius.com or
peoplefinders.com - the "aliases" and former places of residence (if any) they'd list are compiled in the same (or a similar) way.