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.
Ha!... Who would give away the big secret?!:)
In my experience the key is to stay in touch and check in periodically, both personally and professionally with your lead contacts. Especially former clients that were happy with your work. Ask them if they could use you again or refer to their colleagues/peers... Stay on the radar also helps but I do it by participating in conferences, presenting and networking. If you are considered an opinion leader in your topic - people will be asking for help and advise.
Another quick advise - always leave extra value "on the table" and be generous in your own referrals....
I'd be happy to discuss other tricks of the trade if you are interested.
I would say the best way is to ask questions socially / track their support forums / converse with employees to discover a problem they have. If you can demonstrate that you can solve their problem and generate revenue / cut costs to a greater extent than your fees, then you stand a pretty good shot at getting in.
All companies have externally visible problems, if you're willing to look hard enough.