On Tue, 16 Dec 2008 19:48:33 -0800 (PST) in rec.crafts.metalworking,
wrote,
if I did fix it so the part is held, but still allows to adjust
itself, that could help. Maybe if I had the tap head I could almost
just hand hold it in a V block and kind of guide it, while allowing
the tap itself to kind of adjust it slightly as it goes in.

In my experience, that's a bad idea. It's most important to start the
tap as straight as possible. As soon as the tap cuts the first crooked
thread, it will try to use that to guide itself the rest of the way. Not
the hole. #1 cause of broken taps in hand tapping.