Special Snowflake Cover Letter

Did you ever interview for the first position or did you just send your application and then get your rejection? Because if it is the latter there is a really good chance no one remembers you at all (sorry, we go through a lot of resumes). If you never interviewed or anything then you should not address it at all, you should focus completely on this current job and why you are good for it. Act as if you hadn't already applied for another position, send all the information they request again, make it easy on them. You aren't asking them to give you a second look for this position, this is your first time applying for this position, it is basically a blank slate. If you did interview then you should only really address it in passing. I would say something like, "From our previous conversations about the otherposition I could tell that this was a company I really wanted to work for (and then give reasons why)." People apply for multiple positions within the same company all. the. time. no one will think you are desperate if you are applying for two jobs that you are (at least mostly) qualified for and actually want. You only look desperate if you apply for every position they have listed regardless of fit or qualifications.

http://jobhelp.azlibrary.gov/resources/T%20Bar%20Cover%20Letter%20Sample.pdf that I saw on that post yesterday is really good.

I can't quite tell from your question whether you've resubmitted everything, but I would definitely do so again so they don't have to go digging in their archives (aka some random file cabinet somewhere) to find your old writing sample - it's been three months, they might not have it easily at hand. And the way to spin your previous rejection is just as magnetsphere suggests: our conversations about previousposition reinforced my excitement about Company X and the work you are doing. I would love the opportunity to discuss the details of newposition... etc etc. Good luck to you!

Just write a cover letter as though you had never applied to the company. That means if you haven't posted to THIS job, do so. While each position we apply for is deeply personal and contains all our hopes and dreams, sadly, it's quite the reverse for the folks weeding through the resumes. Also, see the http://www.metafilter.com/131829/Sell-Yourself. Most cover letters aren't actually read.

On second read, I agree with magnetsphere--if they never contacted you other than to let you know you were not selected, it is entirely likely you never made it past HR and your prior application was discarded out of hand. If that's the case, don't mention the prior application and just write a regular cover letter.

You don't need to say anything about the prior rejection or assume that it has any bearing on the decision for this position. It's likely they were just looking for a particular qualification that you lacked. They didn't put a black mark next to your name anywhere; as others say it's likely they won't remember your name at all. If they do remember for some reason, what you see as looking desperate to work for a company looks to the company more like being passionate about their company and actually caring about them. It's a good thing. I would suggest using the cover letter to talk 50% about your skills - basically go line by line down the qualifications in the position description and write down why you meet what they ask for in each line, and then additionally mention your academic specialty and any other experience that's hyper-relevant - and 50% about the passion that you discussed here.