Check out openings at nonprofit organizations. They usually don't offer as much salary-wise as big corporations do, but they are a lot more willing to hire people based on their personality, culture fit, and passion for helping their cause, rather than how many years of professional experience they have had.
They are also not bound by the bureaucracy of corporate politics, so if they like you enough, even if you don't meet all of the requirements of the job you apply for, they may still be able to work something out so that they can bring you on board.