blacksmith, the '56 and later 4-speed Hydros were a completely different design from the '55 and earlier. IMO and most other's opinion it was an inferior design. IIRC, the two drivers for the new design were lower tunnel height, and smoother shifts.
Therefore, the fluid coupling was reduced in diameter, which probably reduced efficiency, and a small, secondary coupling was added, which served the function of a clutch pack. Instead of engaging or disengaging a clutch pack, the secondary coupling was filled or emptied of fluid, resulting in extremely smooth, extremely slow shifts. I doubt anyone ever took one of these tranmissions racing. Sounds pretty inefficient, too. I don't recall if the second design retained the mostly mechanical power path in 3rd and 4th.
The first design Hydramatic, when prepped into a B&M "Hydrostick", was the first successful automatic transmission in drag racing, and actually dominated over all other manual and auto transmissions in gasser and probably other classes of drag racing in the early '60s, until superceded by the Torqueflite and Powerglide, I believe.
I forgot what I was going to say