Sadly, a letter offering you a job will not be considered as valid proof of an income source for a new loan. Most lenders want to see that you have been working within the same field for at least two consecutive years. Perhaps not the same employer, but within the same field. They will want proof of income in the form of pay stubs, or if necessary due to self-employment, tax returns.
Programs for New College Grads
Some lenders do have special programs for new college graduates, but these are generally captive lenders (Ford Motor Credit, GMAC, etc), and are only for new cars. Even then, the lender wants to see that you have been working while in college and often requires a cosigner. This person will need to have solid credit and income. Typically a parent would fill this role. Keep in mind that it is a very big responsibility, as the car loan will go on the cosigner’s credit history, and any late payments, defaults, etc, could damage their credit equally.
Another Factor: Money Down
A major area to consider as well is your down payment. How much do you have? Depending on how long you have been looking for work, you may not have enough money down to satisfy a lender.
Despite any hesitation that you may encounter from traditional lenders like banks and credit unions, you may find the financing that you want by searching online. There are specialty lenders who are willing to offer loans to less traditional borrowers. The loan may have a higher interest rate than you would prefer, but the loans are going to be reported to the major reporting agencies, making it easier for you to get a more traditional loan in the future.