Other mistake: Several times, Jennings sees himself being shot on the catwalk, and each time there is a closeup on the flying bullet as it travels toward his chest. But in the following shots, as he falls to the ground, there is never any indication of a wound where the bullet should have hit. The time viewer is always perfectly accurate and bases its predictions on the determined actions of the present, not theories of what actions could take place in the future. There is no way the time viewer could simply be omitting this detail.
Other mistake: When they look at the winning lottery ticket that is hidden under the bird cage the lottery numbers are in the same order they were on the TV but when you buy a ticket the numbers you pick are printed in ascending order, not the random order that they would appear when they appear on live TV. (01:53:30)
Answer: He could have got someone to obtain the key and bring it to him. The same goes for much of the other stuff in the envelope.
Xofer