I stumbled into a lucky situation maybe 5 years ago. I posted a topic on a development forum mentioning my services and linking to a couple of examples. I received a few responses, people wanting work done for a couple hundred dollars, which would basically be a waste of time. Anyway, I get caught up with other projects, and a few months down the road someone sends me a message on AIM, saying they saw my post, and needed a quick project done in the next couple of days. Turns out they're a huge firm in New York with a bunch of Fortune 100 clients. Why on earth they sent me an instant message on some half-assed forum post, I'll never know. I ended up working for them for years and 80% of my work originated from them. It launched my career, and I quickly went from being a freelancer picking up scraps to working on big name projects and networking with tons of great people. All thanks to a random forum post, one of the few I ever posted during that time.
I think that is generally how it works. Big companies don't all operate through ossified HR departments, and even the ones who do still have rogue elements who are responsible for most of the productivity in the company. If anything, I'd probably make it a rule to only work with people who do not try to make me go through HR hoops to work for them.
I don't think of it so much as luck as "having many irons in the fire". None of the opportunities you will end up pursuing will be that good of a chance of coming through, we just kind of delude ourselves into thinking certain ones are better than others. It just becomes a game of stacking up enough low probability opportunities that eventually one of them comes through.
I had my fisrt gig within a month. in that month, I turned down a couple that would have been too low of a rate or bad work.