You 48-52 truck guys are lucky. You have a tool box and a screw to hold down your jack. The Ford engineers must have forgotten about the tools when they redesigned the 53 truck. The 53-55 trucks use the same tools but the jack simply lays on the floor behind the seat. The handle and other tools lay there also in under the seat. There is no tool box. There is simply a piece of upright sheet metal to form a divider behind the seat to the back of the cab. So the tools bounce around a lot.

 

I don't know for sure but I always assumed it goes in the tool box. Since it folds in half it should fit easily.