Annette Alejandro just emerged from bankruptcy and doesn’t have a job, and her car was repossessed last year. Still, after spending her days job hunting, she returns to her apartment in Brooklyn where, in disbelief, she sorts through the piles of credit card and auto loan offers that have come in the mail.

“Even I wouldn’t make a loan to me at this point,” Ms. Alejandro said.

In the depths of the financial crisis, borrowers with tarnished credit like Ms. Alejandro were almost entirely shut out by traditional lenders. It was hard enough for people with stellar credit to get loans. . .