I've known at least a couple of people who entered large firms in different roles (admin support work, such as in human resources) and waited for an opportunity to open up in marketing. Keep going for those marketing roles but be open-minded about other things, as long as the company itself has a marketing department. Or get a job in a completely different work area so you gain employment experience, if you lack it. But, at the same time, develop your marketing skills independently, via social media (nothing contentious of course). You might be able to volunteer your skills in your spare time (eg marketing a local theatre group by helping with their publicity, posters etc). I agree that some employers might see you as 'overqualified' for certain roles, but keep trying.