I'm going to disagree with the the masses here and suggest trying your luck directly with HR.
I once applied for a position and after two phone screens was told I was unsuccessful. I took their feedback on board (I knew where I had gone wrong), and polished up on my skills. Three weeks later I saw the same position was still open. I sent the following email directly to their HR person:
I am writing to re-apply for your position of Systems Engineer.
If you’re having déjà vu right now, that’s because about three weeks
ago you invited me to apply for this position. I spoke to [person1]
and [person2] and my initial contact with [person1] was very positive;
however my performance with [person2] demonstrating my scripting
skills was poor.
I agree that [person2] made the right decision regarding my scripting
skills. So what’s changed?
Since then I have applied dozens of hours of my own time over the past
few weeks getting stuck into my PowerShell skills. They were virtually
non-existent when I first spoke to [person2] (I was only capable of
doing basic tasks like copying TechNet instructions for doing stuff in
Exchange). Although my programming skills are certainly not in the
league of what you would expect from one of your world-class
developers, I have applied Power Shell daily in real-world situations
since then at my day job. Everything that can be done in PowerShell
shall be done in PowerShell has been my mantra.
I realise that PowerShell is just one of many scripting languages I
could have chosen; I chose it based on the fact that your network is
mainly Windows Server (albeit some of them 2003, which does not ship
with PowerShell by default). I hope that the choice of language isn’t
particularly relevant, but rather the fact that I’m hoping to
demonstrate that I’m a very fast learner who can go from 0 to fluent
in a small period of time.
As I mentioned in the first round, I’m actually quite happy in my
current job, so I don’t take rejection hard, but I respect [your
company] so much as to leave my current job, and I truly believe that
I would be an asset to your company.
So with that cringe-worth last paragraph, I ask that you please
re-consider my application for the position. A formal CV is attached.
I had substantially more success - they said it highlighted the fact that I saw a personal flaw and fixed it, and that I came back showed I had the kind of guts that they wanted from a person at their company.