Hey guys, LOOOOOOONNNNGGGGG time, no post.
Anyhoot, I think the answer here lies in, believe it or not, the original comic storyline.
Follow me...
So in DOFP, Kitty is sent back to prevent the death of Senator Kelly. But Kelly is dead at the end of X1 due to Magneto's "mutant making" machine. It is also heavily implied that this machine is responsible for setting off the events that led to Jean becoming Phoenix by the end of X2.
Now, keep in mind that Mystique impersonates Kelly throughout X2, but she is cured in X3, thus making Kelly suddenly "disappear" (a smarter creative team would have perhaps explored -- or at least acknowledged -- this in X3, but I digress...). It would stand to reason that between Magneto's various activities and Mystique impersonating a member of the U.S. Senate, Project Wideawake finally gets the green light, setting the stage for the dystopian future in DOFP. So maybe the future X-Men want to stop Magneto from ever building the machine in the first place, thus saving Kelly and, indirectly, Jean, Scott, and Xavier. Perhaps they tie this to Trask somehow by having the technology that enables Magneto to build his machine related to whatever the Sentinels use to identify mutants?
At any rate, given that Superman Returns was a sequel to a 20+ year old movie, I wouldn't put it past Singer to have Kelly's death via Magneto's device in X1 be a major plot point in DOFP (remember that in SR, Superman and Lois have a kid after one night together in Superman II).
So, to answer the original question in the most roundabout way possible, yes, Logan (even if he's not the time traveler himself, his future self has a role in scheming the time travel plot either way) does try to "save" Jean, albeit as an indirect result of saving Kelly from Magneto's machine.