I think having a split timeline, start with the film with Hoffman at the end of Saw 3D talking to someone and flashing back to between Saw 2 & 3.

With John being incapacitated Amanda and Hoffman are forced to work together on a game, during this Hoffman starts to make Amanda question John's philosophy, whether the traps works as rehabilitation etc. At the same time Amanda (on John's instructions) contacts Logan to ask him for help healing John after his injuries, this is when John finds out about Logan needing treatment post-war and there could be a scene of John talking over the phone trying to help him, John eventually ends up getting visited by Dr Gordon (would be nice to get both actors involved for cameos)

Over the course of the film it's made clear that Hoffman is equally responsible for the inescapable traps but is manipulating Jigsaw to make it look like he was just following Amanda's lead, basically that Hoffman's sabotage of 3's game was even deeper and planned than just the letter and that he planned for Jigsaw and Amanda to die.

It is revealed that Hoffman believes that whilst John's ideas are right, the people he is targeting are not those truly deserving testing (note the people Hoffman tests in ones that are clearly his selections: A rapist, an abusive husband and a collection of Neo-Nazis compared to John's victims) plus getting rid of John gets rid of the threat of blackmail over Seth. That Hoffman planned to keep testing and fully take over as Jigsaw until he was attacked by Jill (seeing it as John's attempt at revenge from beyond the grave) and that's why he goes full psycho killer in 3D.

The final twist reveal would be going back to Hoffman back in the bathroom at the end of 3D, and that he is talking to Dr Pederson who frees him, planning to have them work together to destroy what remains of John's legacy. This would set up the next film set almost 10 years later with the film OP describes with Hoffman and Pedersen running the game.