In summer 2013, Fox will begin this movie continuity-fixing process with the release of The Wolverine. Set after the events in X-Men: The Last Stand, Logan finds himself alone in Japan facing questions about his immortal-like mutant healing ability and thinking about those who have been lost. Obviously the plot of The Wolverine is unknown until it releases in theaters on July 26, 2013, but Millar has already confirmed The Wolverine is indeed the new beginning of Fox's cohesive Marvel movie universe. (Even though it would technically be X-Men: First Class as the new beginning since most of that cast is coming back for the sequel.)
So The Wolverine will set up the events in X-Men: Days of Future Past, but how? While Logan has to deal with the fallout of X-Men: The Last Stand in his solo movie this summer, fans will have to pay attention during The Wolverine to find the puzzle pieces that are sure to be clues for the next X-Men movie.
The Wolverine could have teases like the creation of the Sentinels being planned, or time travel being created. Even though this is a lot of speculation, X-Men: Days of Future Past will indeed have Sentinels and time travel. Millar, himself, has already confirmed both.
So we have time travel and Sentinels, both are big parts of the original comic story arc, and only a few days ago it was also confirmed that the actors who played Iceman, Rogue and Shadowcat will all reprise their original X-Men roles, in Days of Future Past too.
The most important part of all of this is the fact that Shadowcat, a.k.a. Kitty Pryde, will be a part of the new X-Men movie. In the comics, Shadowcat's mind travels back in time to possess her younger self to warn everyone of the pending doom for mutants in the future. An assumption is that, as in the comics version of this story, Sentinels are eradicating the mutant population in the future. To prevent this, things must be done differently in the past. Hence, the time travel by Pryde.
Even though Marvel movies only take a little from the comics and adapt them for the big screen, Shadowcat could still time travel and change the future. And as the future changes, Fox will be creating a brand new, cohesive X-Men universe as they intended. Finally, if Fox does things right and changes time lines via time travel successfully, you can bet Hugh Jackman will be seen in future X-Men movies seeing as how he doesn't age. His cameo in X-Men: First Class proved Wolverine still has and always will look the same—as long as Jackman continues to defy the aging process in real life.
There you have it. Time travel by Kitty Pryde, after the events in X-Men: The Last Stand, to a younger version of her post-X-Men: First Class will change the continuity of the past X-Men movie trilogy to create a cohesive, singular X-Men movie universe for Millar and Fox to play in, expand with more solo character movies and whatever else they may have planned.
UPDATE: It has been brought to our attention that X-Men fans are pretty much 50-50 on saying whether or not X-Men: First Class is a reboot or a prequel. Bottom line, First Class is neither since story lines clash which is why Days of Future Past is an attempt to fix things that are wrong in continuity and allowing for a cohesive X-Men movie universe.