In Logan (2016), Logan decides to help Gabriela's family after she offered him money to help. When he went to the motel Gabriela was staying in, she is already killed. And yet, Logan searched all over the place for her phone and took it - why did he do it?
Logan, no matter how much of a selfish bastard he pretends to be, is still a bleeding heart. He is also rightly paranoid about being found. The phone was a clue to saving the kid, and evidence of how she found him. If Gabriela could find him, so could others. So he needs to find it to protect himself, Xavier, and Laura.