I have a 2 story home in San Antonio that will be available for rent and does allow pets. It is only 5 minutes from UTSA and you have access to the "back way" in with minimal traffic. It is an excellent location and a very nice neighborhood. It has a large fenced back. If you want to contact Dynastar Property Management Inc, (210) 490-9570, speak with Thom or Jana Perdue
This will at least give you an idea of how much rents are, and what you can get for your money. I don't do rentals.. and honestly don't know any really good agents that do... not much help there. Sorry.
Honestly... you can get a mucn nicer home with less money per month if you buy a home. Best buyer's market we may ever see in our liftetimes... I wouldn't waste money on rent while interest rates are sitting in the 3s!