I was pouring over the few books I have, and the 1966 Comet, Falcon, Fairlane and Mustang Shop Manual indicates there should be a Transmission Identification Tag fastened to the top upper right extension housing bolt (pg. 6-10). The tag is supposed give year-month-day and shift.
I just looked, and there's a tag on the bottom right extension housing bolt. It says:
HED-BG
6-E-02
HEF and HEG are supposed to be the codes for the 3.03; 6-cylinder and 8-cylinder respectively. Might HED be the code for the 2.77?
June 21st is a good distance from May 2nd, though. But if the 6-cylinders were a slow mover and the reason for creating the Sprint designation in the first place, and with marketing targeting young teenage women (who likely would have preferred the C-4 automatic, is it reasonable to assume that Ford would have amassed a reasonable queue of 2.77 transmissions; dating back some time before the actual build?