I hope they're shopping him now. Teams in contention should relish the prospect of winning a few extra games in May, June, and July rather than waiting for the last two months of the season.
Of course, I have my doubts that Greene will be much of a contributor to extra wins, because in general a one-inning pitcher -- even a really good one -- rarely makes much difference in his team's record. Such pitchers are more valuable than they used to be, with so many starters not going deep into games, but they're still not nearly as valuable as many people continue to think they are. And analytics have shown that quite clearly.
Nevertheless, there are still some GMs who seriously overvalue relief pitchers, and if one of them offers something really good for Greene at any point of the season, the Tigers should take it and run.
There's a lot of talk about the Braves and their bullpen woes, but there's a good chance that they'll try for Kimbrel after the draft, when he can be signed without his new team losing a draft pick. Of course, given how relatively cheap Greene is in terms of salary, they can certainly afford to have both him and Kimbrel, and they have some every desirable young pieces. Maybe too desirable for them to part with them for a relief pitcher, but we can hope.