"Do big law firms hire during the summer after 3Ls have graduated? Should I keep looking while studying for the bar?"
Yes, big law firms are probably hiring if the right people come along. It's much more difficult to get a job as a 3L. After your 3L if you haven't worked in a big firm, but I would keep looking. If you don't have a job in a big law firm after the summer, if you haven't had one in the summer, then it does become much more difficult to get one, but just the other thing to remember is that your career is a long race. You don't need to get a job right away, and you also do not necessarily need to start in a big firm to get to one.