No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Improving her score may help, but student loan lenders also look at income and, if they review the file manually, will still see that the student has limited experience with credit on their own. It's fairly unusual for a undergraduate with no income of their own to get private loans with no co-signer. An older student with their own income has a much better chance.