Yes, the 4 speed Hydramatic was an option in 56. The kickdown is controlled by a mechanical linkage from the carburetor to the transmission, which also controls shift points. There is a small lever at the top of the driver's side of the transmission near the shift linkage which moves as the throttle is opened and closed. Near full throttle, the lever should reach its full travel and cause a downshift from 4th. to 3rd speed. The linkage is adjustable for length, so you can experiment with lengthening or shortening it.
A Hydramatic is a fairly complex transmission which is difficult to repair, and repair parts are somewhat expensive. One company that deals in obsolete transmission parts is Fatsco in New Jersey. I've done business with them for over 30 years. https://www.fatsco.net/
Finding someone in France with experience repairing that transmission will probably be more difficult than finding the parts! Good luck!