We often focus on the $38,500 cap on sub/unsub Stafford loans, but there's another cap that's going to hit you.
Most of these educational lenders will only provide loans up to your Cost of Attendance. That includes all forms of financial aid -- Stafford loans, Perkins loans, AND private loans.
The cost of attendance at my state school is around $29000. Even though I haven't drawn the max in Stafford unsub loans, I can't get the extra $9k, even through private loans. Kind of bites, but very few educational lenders will offer you private loans above your COA (T.H.E. will not).
For any more money, you're going to have to head to a bank and take out a traditional loan. Much higher interest rates, no repayment breaks/perks, and almost certain that a co-signer is required. You'd be lucky to even find one that would let you defer payments until graduation.
This issue doesn't hit a lot of people because most med school tuition rates push the cost of attendance way above the Stafford cap. But in your case it just might not.
If you have kids, the school can raise your COA to include day care costs, medical bills, new computer. Kind of dependent on how much you can strong-arm the financial aid office.