I think you could safely leave Corpus at 6:30 a.m. and comfortably make your flight, especially if you are checked in already. It'll take less than 10 minutes once you drop your car at Hertz to get to the terminal. At 10:45 a.m. or so, the security line at HOU is usually very short.

Your Hertz rental car should come equipped with PlatePass, which will allow you to drive through the EZ Tag lanes, with your credit card billed after the fact. There is, however, a daily surcharge for using PlatePass. Personally, I tend to bring a handful of quarters for the toll road and pay as I go.