I been using a TTS adapted Procunier 1E reversing tap head for tapping 10s of thousands of 4-40 blind holes with great success. BTW I also have Tapmatics but never use them. The Procunier with cushioned double cone clutch just works better.
The Procunier also taps 2-56 blind holes easily.
Tapping 2-56 on Tormach video by miltons_stuff - Photobucket