If you end up going to UT, they have orientation sessions during the summer that last a week. The fraternities hold events during these sessions. They all hold "late nights", which are parties either at the house or at a place they've rented downtown, and during some sessions, big theme parties are held, this is where you can make yourself known (in a good way) to the houses. These are open events and if you're rushing, not just some random guy, you will be welcome (unless you make an a** of yourself

).
As to fall rush, I don't think they really do it (at least my son's fraternity doesn't), bids are extended before school starts. I know they don't have a structured rush like Ole Miss where all the houses hold events on some schedule.
You won't be the only new guy from out of town going through rush. My son's roommate is from Indiana and didn't know a soul, and he pledged. He's just an awesome guy who came across very well during his orientation week, and he fit in with the house.
Like I said before, in my son's pledge class there is a guy from England, his roommate from Indiana, and also a guy from Chicago (and Indiana and Chicago might as well be from Uganda

)
While it is true that some pledge classes (including my son's) will have 15 or more guys from the same highschool, there are other guys who pledge too. I think this year's pledge class at my son's fraternity had 50+ guys, so there is space for new people.
FYI - If you are planning on visiting in May, you will have made your choice already - May 1 is the deadline for accepting admission and paying your enrollment deposit at UT.