Bird Dog states that September 13, 1956 is a Friday. In actuality, it was a Thursday.
Near the beginning of the episode, to establish that Sam is in the 1950s - The Howdy Doody Show (1947) (AKA The Howdy Doody Show) is shown playing on the television set. September 13th, 1956, the date that this episode takes place, fell on a Thursday. In 1956, this show was broadcast on Saturday mornings at 10 AM and would not have been on the TV that particular morning.
When Sam awakes in Tom's body, it is 5:00 AM on 9/13/1956, and the sun has risen. However, on that date, sunrise near Edwards AFB wasn't until 6:32 AM PDT.
Early on, Al is seen by Sam in the back of the cargo plane with his clothing flapping in the wind. Though, to Al, his surroundings are nothing but a hologram, and thus, nothing around him should have any physical effect on him - including wind.
The shirt on Sam's "reflection" as ballplayer Tim Fox is not reversed. You can clearly see the buttons and capital "B" (for "Bombers") on the right-hand side.
When Sam is co-piloting the bomber and the pilot leaves his seat, the image of the plane from outside is backwards. The words "US AIR FORCE" on the nose are printed backwards.
When Sam looks into the mirror to see the actor that he has leaped into he is shocked and moves back, but the actor opposite him is delayed by about half a second.
As one watches the episode, the scenes that show the actual planes flying, one can tell that stock footage from old film reels is being used, as the quality is not as good as the rest of the episode.
When Sam leaps, he finds himself at the controls of an aircraft he has no ability to fly. Later, he is shown as the copilot of a B-29, still unable to fly. As the copilot he would have had to take part in lengthy and involved checklists in order to start the engines and pre-flight the aircraft. With no knowledge of how to actually fly a plane, there is no way he could have completed these tasks without others in the plane realizing he couldn't fly.