Here are all the pieces. The thumbscrew is a #8-32 that I turned the end down to 1/8" on so it could ride in the 0.135" wide groove. The diameter of the plunger, as I'm calling it, is 3/8". The OD of the fixed section that will go in the tailstock drill chuck is 1/2" with a 3/4" raised section on the front. I don't remember the pieces' lengths, but the OAL is right at 5". The piece to the left of the spring sits down in the spring and is bored to accept the turned down sections of the plunger. The plunger has 1" of travel, which should be plenty to get a tap started far enough to not need the guide.
(edit: The bigger section of the barrel is actually 0.800" in case it matters to anyone. It had to be thick enough to leave some threads after I milled a 3/8" wide flat for the screw to rest against.)
Something I learned today was that the pointed back end of taps are about as non-standard as trees! I have taps that have included angles of 72 degrees all the way up to 127 degrees! Some are actually pointed. Others have a flat almost 1/4" wide. Makes it pretty difficult to make a tool that does it all. I'll have to make another plunger that will work with some of the taps with wider flat areas. This guide should handle pointed up to a flat of about 0.090" wide. Even at that, care will have to be taken to not pull the tap out of the opening. That opening, BTW, was made by a #2 center drill.