Find a place where you can talk to someone in power (i.e., not just HR) and demonstrate your passion for what the organization does (it helps if you actually have some).

I wiggled my way into my first programming job because I really liked what they did, and made it clear to them. My boss later told me that the code samples I submitted weren't very good, but anyone can learn to write code--He hired me because I really demonstrated a desire to do the work.