Given the timeline—that Logan survived a Jigsaw game 10 years before, then worked with him in some capacity that may or may not have involved making the reverse bear trap Amanda's later put into, and has since eked out a semi-normal life with his daughter—it's something of a mystery as to why he'd start re-creating Jigsaw's early unknown games now all of a sudden. The answer, it seems, is that revenge is a dish best served cold. Although he's advised by Jigsaw not to let a thirst for vengeance control his behaviors in the games, his new effort seems to be driven by the sole desire to punish Halloran for letting terrible people off easy.
The man who'd ended Logan's wife, it turns out, was Edgar Munsen, who's killed in the beginning for carrying a remote that he's hurriedly retrieved from a spot marked with X. He insists he's been told to choose whether he lives or five others die, and he's immediately gunned down by police (with one bullet lodged in his chest cavity by Logan, secretly snipering). He survives the wounds, but ends up in a coma. It doesn't take long before he's snatched from his hospital bed and winds up in John Kramer's coffin, which is exhumed to quell the public's fear about the possibility that Jigsaw is still alive.
There are other people in Halloran's history who've also been able to bribe their way to freedom, only to hurt more innocent people. One such victim was an eight-year-old who ended up on Logan's table, and, since that's roughly how old his own daughter is, it seems to have set him off on a mission to end Halloran and whomever else he kept out of prison before they can hurt anyone else.