How can I look discretely for a job within my industry without the
word going out about this?

I'm in a rather small industry group and many players know me, as I do
Business Development for my company. I don't want to hurt the company
by making public that I'm looking for a job. As soon as I would
contact some of the players in the other companies, this would become
public,

Is there a headhunter (personnel placement) that specializes in your
industry or field?  A good one would handle your name discretely.