Is it inappropriate to ask for a 2nd interview after being rejected in the 1st round?

I was interviewing for a school counseling job and I didn't make it into the 2nd round of interviews. I am really upset because I am an internal candidate and the principal, HR Staff, and the interview team knew me. Heck, I got recommendations from half the interview team. I still didn't make it to the 2nd round. I am so irate. I was thinking of asking for a 2nd interview. I was going to ask the H.R. manager, the head counselor at the school and the principal. Good idea? Worth a shot. Or will this hurt me getting a job as a school counselor in the school district in the future?