There is a certain state agency that I would really like to work for. I happen to have a fair number of dealings with a particular office of their’s, and last week, during conversation with a paralegal there, she mentioned that she would be leaving soon so she had been swamped lately since she was getting ready to transition out at the end of the month.
At which point I wondered if there is a polite way to ask an almost-total stranger if you can have their job.
Probably not, right?
Especially not when you’ve been calling them with all kinds of dumb questions and generally sounding like a moron who doesn’t know anything about anything?
But you better believe that I’m stalking the agency website, waiting for her job to be listed. Even if it is a paralegal position, it’s probably a really good way to get a foot in the door of the agency. Or at least get that fancypants health insurance stuff.
Any other suggestions on how to get inside information or a foot through the door?
Vado Porro - Go Further 
I actually don’t think it’s inappropriate at all to ask a near-stranger for their job. If she felt comfortable telling you she was leaving, I think it would be perfectly fine for you to call her and say, “Last week, you mentioned that you would be leaving soon, and I’m actually really interested in working for CompanyName. Could you let me know if it would be possible to apply, and how to go about that?”
Asking questions doesn’t make you a moron; it makes you someone who wants to do things correctly, and also someone who is a little more familiar with their processes. Plus, having her send in your resume, instead of just randomly applying on the website, gives you a much better chance at getting an interview. The only thing that would hold me back from approaching her would be if it was really likely to get back to my boss.
I say – go for it!
The great thing about being under-employed is that my boss is totally supportive of me searching for new opportunities and I already mentioned that I wanted this job :).
I think this is perfect advice. Ask her for suggestions — if there is anything that helped her do her job really well, what she will miss most of all, and the logistics of applying.