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OtherJen, do you do anything special in terms of how you use LinkedIn/present yourself on the site? I've come to realize that there's a lot of undocumented knowledge floating around about what works and what doesn't work on LinkedIn and being based in academia I'm exposed to almost none of it.
When I was still relatively new to my current career (about 15 years ago), I joined LinkedIn and was very active with it. I never set my account to looking for jobs, yet I've had two contracts arise from it and hundreds more that I'm not interested in. I attribute my success on it to creating groups. I was no expert, but no one else had created these very niche, subset groups. I use a few software products that are popular in my industry, so I started the "[product] Working Group". I'm not an expert, but I wanted to get better and I wanted to have easy access to the people who are experts with that product. Then I used attendee lists from conferences I had attended that are related to that software product, or at least to the industry, and emailed anybody I could find to personally invite them to join the Linked In Group. (LinkedIn has options for inviting people...you just supply their email address).
Then I did the same thing for another software product that I have actually never even used, but that is common in my industry. Then I did the same with a certification that is used in my industry as a "study group". Great success!
Not only did I get to network with all the leaders in the field and get free help when I ran into a stumbling block, but when companies/recruiters are searching for someone in my field, they use those product names as keyword searches and I come to the top of the list because not only am I associated with that group, I'm the OWNER/MANAGER.
I also make it a point, when I receive what looks like an interesting job lead, even if I'm not interested, to send the lead to some other contact in one of these groups. It's the easiest method of networking....think about how you feel when a friend of yours recommends you for a job. I do this all the time, and now all of those people think about me when they need to fill a vacancy.