How to tap a deep hole?

Sounds like you need a long 3/8-16 #3 tap, also known as a "bottoming tap." A given size of most taps come in -1, -2, and -3. The -3 have nearly square threads on the bottom, with only a slight taper to get them started. They're designed to be used after starting the hole with a regular tap. You car order sets in all three tapers from places like MSC, Fastenal or McMaster-Carr. Or find an old school machinist.

McMaster-Carr