Last time we noted that if Amy jumps from Wednesday to Saturday, she cannot be killed on Friday, and that it is highly improbable that her friend Carol would come to see her and be killed on Friday at her apartment. Yet for Amy to believe Wells' fantastic tale, she must find the paper on the table in the museum; and for that paper to be of use to them in their efforts to capture Stevenson, there must be a current story recounting the deaths that had occurred. It is likely that such a story would be on the front page of Saturday's paper only if Stevenson killed someone late Friday or very early Saturday.
If he does not, Wells and Amy have no clues to lead them to Stevenson, and we are stymied as to how they will find him--but that he has left a note demanding the key and threatening Amy's life. Delores Marx will be killed on Thursday, and then on Friday Stevenson will face Wells and fight over Amy, probably killing (or at least threatening) Carol. The police will not be questioning Wells, because he did not report the murder of Delores and so is not suspected. One way or another Wells defeats Stevenson. Presumably Carol is killed, someone reports the murder, and it is reported as Amy, putting it in the paper for the version we see.
If (in the original history) Stevenson kills someone else on Friday, we have a front page headline story reporting it, along with the names of the previous victims including Delores killed in McLaren Park; but there is a different Friday victim. Wells calls in the Marx murder, is brought in and held for questioning, and shows the police the paper in which there is a murder predicted, but not Amy's. The police do not believe him anyway, and continue to hold him until such time as Amy's murder is reported, at which point they release him. Stevenson never intends to kill Amy; he intends to use her as a bargaining chip. Amy already invited Carol, so it is Carol who is killed. The police might threaten Wells with interference with a police investigation, since the incorrect identification of the victim (which they did not pursue anyway, as they thought Wells was the murderer) confirms their belief that it is a novelty newspaper. Wells wonders why the paper was wrong, but again he manages to defeat Stevenson and save Amy, then leaves for the past. She probably goes with him.
Of course, if they prevent the murder of Delores Marx, it will not be reported in the paper; probably nothing will be in the paper, if Stevenson is caught on Thursday, and that means they will not have the information needed to prevent the murder, and we have an infinity loop However, the only difference in the timeline in which someone, not Amy, is identified as the Friday victim is that there is less stress on the twosome to solve this quickly. It is possible that the absence of Amy and Wells might (butterfly effect) cause Stevenson to kill someone else on Thursday night, but it does not appear as if they had any significant impact on him Thursday so this is unlikely.
Ultimately, then, we have the history in which having visited the future and returned, Wells and Amy cause the existence of the headline reporting her murder, which turns out to be Carol. That becomes the paper they find when they travel from Wednesday night to Saturday night, and the paper tells them of Amy's murder, which gives us our final version of the history we see. At the end, on Friday, he takes Amy back to 1893.
The movie is finished, but we are not. That return trip creates an entirely new set of problems, a new history to be examined.