It is a good idea to reach out to your connections at the places you apply. You want to
- let them know that you are applying,
- express your enthusiasm about the place,
- remind them briefly about your research,
- ask them to bring the application to the attention of their colleagues on the search committee.
Dear Susan [or Dr. Williams, if you are not on a first-name basis],
I was intrigued to see the recent advertisement for a faculty position in the area of cosmological neuroeconomics in your department. While I have a year left on my postdoc, your program matches my interests so well that I did not want to pass up the opportunity to apply. Moreover, I've always loved New England and would leap at a chance to move back to the area.
I know we had a chance to talk about my work on hypothetical ultimatum games in extraterrestrial populations at the San Francisco meeting last summer, but you may not know that I've started to move into the area of inflationary neuroimaging. I have [list a few findings or directions here]. As such, I feel that I would bring an important set of skills to your department that is already so strong in this area.
If you think that I might be a good fit for your department, I'd be most grateful if you would bring my application to the attention of your colleagues on the search committee.
Best wishes and see you at AACNE in Chicago next summer.
You are not asking for a special favor or doing anything inappropriate. Instead, you are doing your colleague a favor by letting her know that your application is in the pool, and you are doing their department a favor by helping them sort through the hundreds of applications that a department may receive.