After Jack's last return to the past, he does five significant things, all of which we have already mentioned: He gives Doctor Becker the names, tells Doctor Lorenson how to cure Babak, delivers a letter to Jean, bangs his head, and gets back in the drawer. This changes every future history, and in many cases it gives us the future we see in the film. It gives us all the parts about Jack dying on January first of a head wound.
However, that means that in any history in which Jack receives the head wound Jean got the letter; and if Jean got the letter she didn't die, and Jackie is not the waitress but the hospital worker. This creates a much more complicated problem, but it means that the scenes we actually see in the movie could never have happened as we see them: by the time Jackie knows that Jack died on New Year's Day, he has already given the letter to Jean, and nothing else he does or does not do will change whether Jean reads the letter and changes her life so that she lives. It certainly appears that sending the letter fixed Jackie's life, and probably it did; but it did so long before she began trying to find out what happened to Sargeant Jack Sparks.
That raises another serious problem. We can certainly accept that in his first trips to the future, that is, the first time he left from the drawer and the second time he left from the drawer in their original histories, he discovered a world in which Jean had died and Jackie was struggling as a waitress drinking herself toward death. However, once he sends that letter he changes Jackie's life, not merely when he arrives on his last trip but also when he arrives on the other three. Consider: if the letter Jack sends alters Jackie's life such that she works at the hospital instead of the diner, then the first time he arrived she worked at the hospital rather than the diner; if the letter means Jean never died, then the first time he arrived Jean had not died. He has altered the future history of the world in a way that undoes the bad events he witnessed. He will never meet the self-destructive waitress version of Jackie, never learn that Jean died in a fire from her own cigarette when she passed out from her own drinking--and, ultimately, never write the letter warning her of these things. He has undone the information that is the basis for what he does to undo it. He has created an infinity loop. It stretches from the first moment he traveled to the future (because this is where he gets, or in this case does not get, his information about the future) to the moment he leaves for the past on his final return (because this is the last moment of that unaltered future which he is about to change).
This of course could work under multiple dimension theory, because Jack could have visited a world in which Jean died and then created a history in another world in which she survives; however, he still would not be able to find a world in which he died of a head wound on January first and did not deliver the letter, so that remains a problem. Further, we have already seen that other aspects of the film do not work under that theory. Further, in this world it must still be that the Jack in the asylum has traveled to the future, and thus that this Jackie would recognize this Jack as that one--it appears that she has no idea who he is when she last emerges from the diner, and that means that she never met him before, that no version of him ever traveled to this future. It is not rational.
Overall it was an enjoyable movie, and if you don't think about it you might even believe it to be plausible. However, it is inconsistent in its handling of the changes it makes, some things seeming to work under fixed time and others under multiple dimension theory. If the Babak problem could be resolved, it might all fit under replacement theory with an infinity loop ending, but Jack could not then arrive in the future to meet Jackie so we would have no ending to the story.
We do have an ending for our analysis, however. It was an interesting, convoluted, and challenging film, but in the end it fails as a time travel story.