Go to (and preferably speak at) the same user groups and conferences that people from that company go to. Hack on the open source projects that the company uses internally. You will soon have friends in that company, and probably 10 others.
This is assuming that they are using open source projects internally and that you A. Know what they are using and B. That they care about who maintains them rather than treating them as a black box that does stuff for free (how most open source projects are used most of the time in practice).
Also tech conferences seem to be a pretty expensive way (at least in terms of travel) for hoping to bump into someone.
At some point, if the company is not making efforts to reach out to better programmers (by actually looking up the name behind the open source project and showing a strongly present figure in tech conferences) then it's their loss, no?
If a company only trust "recruiters" to select (or pre-screen) candidates for them, then this says a lot about their way of managing and interacting with "our" kind of programmers (by that I mean people who read HN, are enthused by cool innovative projects, write blogs, publish open source code, ...)
If the company doesn't help you bypass the recruiter, maybe you should bypass them altogether?
"This is assuming that they are using open source projects internally and that you A. Know what they are using and B. That they care about who maintains them rather than treating them as a black box that does stuff for free"
Then turn it around and look for companies based on the open source products that they use. There are plenty of companies that are heavy users of open source software, and don't hide that fact. You might have to move or something, but that's life.
"Also tech conferences seem to be a pretty expensive way"
There are many cheap conferences, or free exhibit hall passes for early-bird registration (usually allows BoF sessions, as well, and maybe parties); and user groups are ordinarily free (maybe you chip in for dinner or something). There are conferences or user groups in many areas about something, and maybe you can just drive. Often, you can share a room with another attendee (ask on some mailing list related to the conference) to save on lodging. If you have developed something or have some kind of speaking experience, the conference may pay for some or all of the travel expenses, and may waive conference pass. There are even some open source projects that have money waiting for anyone who wants to bootstrap a user group in their town.
And these things are not theory. I have done absolutely everything I have suggested to you here (except trying to bootstrap a local user group): get free conference exhibit pass, drive to conference, share room, get started contributing, get major patches accepted, and get jobs through community connections. And I still do it, and I see many people rising up through very similar means that had little in the way of marketable talents a while ago.
Disclaimer: Nothing about this happens overnight. It takes a long time. There are no magical steps where I can say "do these things and you will get this". You have to look out for the opportunities on your own. Start attacking your own reasons why something can't be done, knocking them down one by one.
I do in theory like the idea of speaking a tech conference. I am just worried about the level of experience that would be required to do so.
I don't want to stand up in front of X number of people and tell them stuff they already know.
I live in a place that is very very far from being a "tech center" in the way that SV is but there are a few user groups around (mainly PHP groups, LUGs etc). I should probably check them out.
I did go to a local LUG once but that was about 5 years ago and mostly consisted of unemployed neck beards grumbling into their pints of bitter. Hopefully things have changed since.
"I don't want to stand up in front of X number of people and tell them stuff they already know."
A topic that is always safe is "I used technologies X, Y, and Z to build this thing, and here's what I learned along the way". It's usually a little more entertaining because it's a story (often with a few funny anecdotes); and you're admitting that you didn't do everything perfectly from the start, so being wrong somewhere during the presentation isn't a deficit. Being wrong really only matters in a persuasive talk.
(Aside: it's pretty well established that teaching is one of the most effective ways to learn deeply.)
I'm sure you can find better speaking advice elsewhere, but one other thing I should mention: especially at user groups, a talk is just to start a conversation, which will often be held at a bar later. So don't worry about concluding anything in a profound way.
As for ___location, then yes, living within driving distance of the SF bay area helped me a lot. But I didn't begin speaking until I moved to Portland, Oregon, where I regularly attended a user group. I'd like to think that a lot of other areas have something to offer as well, but I'm sure it takes more searching.