ubi, a good question. If you are allowed to search for a new job, then why can’t someone help you in that? Maybe the difference is who is Jewish in various scenarios. Would result be different if the babysitter were Jewish? If you have a chance to give a Jewish babysitter a better job, is this wrong?
But re:headhunter, even if it is mutar, you may not want to do that job yourself. Similar to meter-maids – they do a necessary job, but who wants the ayn hara!? (I once put a quarter into someone’s meter when the metermaid was coming and she gave me an ayn hara back!).