I don't think that's the way it works. Even though we've DFA'd Hanley he's still going to get all of this year's salary.
The best way we can get salary relief is if the Sox trade him. In that case the team that takes him in the trade is responsible for all of his remaining salary. That's why nobody is going to trade for him.
OTOH, the Sox have a week to trade him and after that week Hanley becomes a Free Agent and able to sign with any team without the Sox being involved. The Sox are then responsible for Hanley's entire 2018 salary minus whatever agreement Hanley might work out with another team.
In that situation Hanley has no incentive to drive up his price because he's going to get the same pay whether it comes from his new team or the Sox. Also the receiving team has no incentive to pay him more than the ML minimum because the Sox are going to pay whatever portion of Hanley's salary the receiving team pays. So he most likely will end up with the new team paying him the ML minimum and the Sox making up the difference.
The best case scenario for the Sox (other than a trade - which most likely won't happen) is for two or more teams to get into a bidding war for him, drive his price up and thereby drive the Sox commitment down. But don't hold your breath for anything meaningful. The savings come when/(if?) the Sox don't have to pay him for 2019.
BTW, how's your softball team doing?