It depends really on your credit situation... Some banks will be willing to give you a personal loan (instead of a car loan) for the amount you are asking and "secure" it against the car you are buying, if you are not co-signed or have a guarantor.

It maybe possible to get an unsecured (or secured, if you have some other assets like a house) line of credit, which is often more flexible than a conventional loan.

Shop around, don't be locked into a single bank mindset, thinking that your loan/creditline must be done at the same bank your bank account is. Banks are businesses too, and if your "home" bank doesn't meet your needs, there are several others who I'd bet would be happy to have you bring your business to.