Check your vehicle's liability insurance - does it cover U.S. rentals?
Do any of your credit cards?
If your auto policy covers U.S. rentals - make sure the liability coverage is at minimum $2,000,000 as hamburger mentioned (higher is better considering that you'll be in Texas and Chicago - Cook County is one of the hell holes for insurance litigation)
Again, if your auto policy doesn't cover U.S. rentals, go over that rental agreement for liability coverage. It is crucial.
Make sure the rental company knows where the vehicle will be travelling - some have travel exclusions (brendalee and I once got one that wouldn't let us go closer to NYC than 50 kilometres
) - if you run into some kind of problem and you're somewhere they don't expect you to be, there can be difficulties i.e. the liability coverage won't actually cover you.