Should you thank first-round interviewers for advancing you?

stackerquad's picture
Rank: Monkey | 43

If I get second-rounds/final-rounds, I usually email my first-round interviewers (if they're bankers and not HR) informing them I've progressed, and ask their advice for final rounds. Should I also put a line in the email thanking them for recommending me for the next round in the first place?

EDIT: Would it be any different if the first-rounds were phone interview vs. in person?