Let's look at how this works...
When you completed the FAFSA, you submitted that information to the US Department of Education. The FAFSA is their form, and they're the organization that's responsible for processing it.
The Department does two things with your form - they determine whether you are eligible for Federal Student Aid - and if so, they take all of the information that you have provided, and fold it into a giant formula that calculates the financial aid index score known as the EFC.
That's it. At this point, the Department of Education is finished with your FAFSA.
Meanwhile, your EFC score gets sent to your school. It's the financial aid office there that is responsible for putting together a financial aid package for you. Your EFC score is used to determine your financial aid need, and it's also used to determine what forms of financial aid you are eligible for.
When your school has reviewed your EFC, they'll tell you what kinds of assistance they can offer in an "aid offer letter". You'll get one of these every year. The letter may contain any number of different types of aid - loans, grants, scholarships, and/or the Federal Work Study program.
The aid offer that you'll find in that letter represents everything that your school has to offer you. Sometimes the offer may provide everything that you need for all of your educational expenses, but more commonly, your financial aid is exactly that - "aid" - as in "assistance".
If you find yourself short of what you need for school, your other options are out-of-pocket contributions from yourself and/or your parents, private scholarships, and private loans. Your school can't really help you with any of those things, other than maybe suggesting some contacts for you.
The best thing I can recommend to you is a supplementary program to the Stafford loan - another government lending program called the PLUS loan. PLUS stands for Parents' Loan for Undergraduate Students, and as the name suggests, only your parents can take out a PLUS loan. Like the Stafford, the PLUS is another low-cost borrowing option, but there are some key differences.
Most importantly, the PLUS is a credit-based loan, which means that your parents could be rejected for the loan if they have bankruptcies, foreclosures or very overdue bills on their credit reports. The good news, however, is that PLUS lenders are far more lenient than most other lenders.
The other option for supplemental borrowing is private loans, but you're going to need a cosigner if you hope to have your application accepted. Your school can recommend some major educational lenders - look for familiar names like Chase, Wells Fargo and Sallie Mae.
I hope that helps you - good luck!