You shouldn't limit your search to banks that you have business with already. Many times the cheapest rates can be found at a local credit union. Check for some in your area and see what the membership requirements are. You might be surprised at how easy it is to join and get a great rate.
Also, looking at used late-model cars right now should not be your only focus depending on what type of rate you are able to get. MANY manufacturers are offering low or zero rate financing and cash incentives which can often equate to the same payment as a lower price for a used car at a higher rate. Just something to think about. If you aren't able to do the calcs yourself check out one of the sites below or one like it to see what your payment tolerance is like with different purchase prices and rates.
One last thing, ALWAYS negotiate the OTD (out the door) price. Dealers always negotiate the vehicle price then you get hit with taxes, title, processing, other bs charges.
I work in Finance so feel free to PM me if you have any specific questions.
Good luck in your search.
James
http://www.cars.com/go/advice/financing/calc/loanCalc.jsp?mode=fullhttp://www.bankrate.com/calculators/auto/auto-loan-calculator.aspx