I guess I should have been more clear in my post. I don't have any degree.
If you haven't yet, one thing that would be worth doing is working with a recruiter. There's plenty of bad ones out there, of course, but a good one will have direct access to the people hiring. If you're willing to work on-site, even better, as that will significantly reduce the number of people you're competing against for attention.
I hope you find some of this useful, and either way, I wish you luck!