Another option is a tapping block. holes sized to just allow taps to slip through, faces are perpendicular to the holes thus holding the taps straight.
This was one of the many tools I had to build back in machine tool school. This one is sized up to 1/2" taps, could easily build a smaller one for the little stuff. I had to make it on a jig borer but I think if you were careful and took your time you could do it on a regular mill as well.
Was made from O-1 and hardened IIRC, but for the small taps you guys are using a softer steel would probably work fine