Two additions to my shop tooling.

[stevec]

Starting with a 4 flute plug tap is also problematic. I have was taught that to tap blind hole (not your case, ML) as deep as possible, one should start with a starting tap, second stage with the plug tap and finish with the bottoming tap.
Although, I admit, I have gone from starting to bottoming taps without using the plug tap, it's a "do as I say, not as I do" sorta thing.