you should be looking at private lenders that offer graduate-level or law school loans. i, personally, went with citibank.
i wouldn't worry too much about timing; to apply, you'll have to tell the lender where you're going to school, how much you need, whether you're using a co-signor, etc. if approved, they'll send you a promissory note that you have to sign and return to them. when they get that back, they'll contact your school's financial aid department to confirm that you're requesting an approved amount. at my school, once that amount was approved by the financial aid dept, the lender sent the check directly to them, who then cut me a 'refund' for the amount that i had coming to me (total amount of loan - school fees).
start shopping around for interest rates, you may be surprised at differences between lenders.