I've heard some folks find ways to get in aggressively, while others talk about networking and "staying on the radar" until the big company choses you (because you were friendly and stayed in touch meaningfully on LinkedIn or something along those lines).
Recently, I've hit a "tipping point" where I have enough content (blog posts, podcasts, interviews, etc...) that companies now regularly seek me out for advice.
"To be known, you must teach" - Nathan Barry
After I wrote "This is a web page" (http://justinjackson.ca/words.html) I was contacted by VPs at Coca-Cola and Google. They weren't offering me work: they just wanted to say they enjoyed my writing.
Writing, and amplifying your content, is one way to get your foot in the door.