That has happened to me only once with my FE... when I received it. After unjamming it and replacing the battery the jam never happened again. I'd start looking into the possiblility that the linkage that jams is either bent or unlubricated -- making it more prone to jamming than it should be.
I don't know what, if any, effect turning the camera on and off has on this jam. I have a MD-11 on mine and recently found out that both the MD and the camera batteries will drain if the MD is not switched off after use. But that is just an interesting fact that might have absolutely nothing to do with your jam problem.