The final solution consisted of a socket in the frame for spindle of the second pinion, a short spindle, and the threaded bolt screwed into the end of the frame.  This bolt was ultimately designed to push short spindle into the socket on one end and provide a means for attaching wooden handle on the other end.  The parts are shown in the image below.  Following are images of assembly and the final look of the handle mounting bolt.

Sometime in early 1950s the 1938 No. 2 Hand Drill received new design of the whole second pinion/handle mounting assembly.  It is important to note that 1950s is my own estimated time period.  There is not documentation that I am aware of that would precisely establish the timing of this change. 

However, in late 1940s and early 1950s the Millers Falls Co. developed various new lines of products targeted for mass market and I believe these changes were dictated by the need to shorten production time and reduce cost.  Improvements to the No. 2 drill would be natural focus and the new design for second pinion/handle mount assembly was one of them.

The image below shows new assembly.