If intending to 'travel around' be sure to check the school's contract- I recall a clause in a Florida school contract agreeing that you would not hire an a/c from another school or FBO whilst there on your M1 visa. Whether this was just the particular school trying to protect their business or something related to visa terms or TSA rules, I don't know... Perhaps it may contravene your visa issuance that you will be checked out by an organisation other than your flight school, as that is essentially seen as dual training
On another note, I had intended hiring an aircraft solo for x-country trips after my PPL training in USA, but I must say, after bare minimum training, diving into that might not be such a good idea, I'd want an instructor or experienced PPL with me.