I had not remembered, but early in the film, the crew gets into chambers prior to deceleration. The chief hurries up the men, asking them rhetorically, "Ya wanna bounce through this one?"
I think this is very sophisticated for 1956 and I am betting not only is this the first film that has a special procedure for the transition below the speed of light (and perhaps the same sort of thing when the ship is preparing to accelerate) but also the only film for at least a decade which has this sort of thing. Star Trek does of course distinguish between warp and sub-light but can't think of any other film/tv show that does between 1956 and STOS.
The Wikipedia article says this is the first film in which humans create an FTL ship, but I guess in whatever film (can't think of what that film would be) has a FTL alien ship there could be details as in FP.