Do you already work at this company? And, is the new/second position in the same group/department as the first?
I handling the hiring logistics for my department (job reqs, screen resumes, schedule interviews, interview, etc) in a big company, and we don't see any resumes unless they are sent to us by the system, or emailed to us by a friend/colleague/parent/whatever.
So, I would apply for the second position and not even mention it to the HR person unless you're sure its through them (it might not even be the same HR person depending on the size of the company).
Anyway, its rough out there for candidates, so its not unusual to see the same person apply for several jobs - I don't particularly care as long as they are qualified. The college kid who's previous experience was working at Quiznos applying for 8 different Sr. Chief Engineer positions has me roll my eyes and disregard them.