I guess I just have a hard time seeing Giles sign for markedly below his perceived FA value. The way I see it is this: the Jays decide they have $30-40M to spend on an elite reliever. They have three possible ways to spend it:
a) spend it now on Giles on a FA value contract
spend it now on Giles at a rate less than the value of a FA contract (Giles accepts less money than he'd expect to earn in the offseason)
c) wait until the offseason to make that choice
I don't see a realistic situation in which c) isn't the ideal avenue. Clearly a) is a poor choice: you're nuking his trade value, and opening yourself up to long-term risk by extending him a year before you have to make that commitment. would require a pretty team-friendly deal. The team might think Giles is going to get a Will Smith contract in the offseason if he's healthy (3/39). At what point does it even make sense to make the long-term commitment now, rather than wait and see what happens in the offseason? If the team has 40M earmarked for a late inning reliever, they'd certainly be able to land an elite arm in the offseason: Giles, Betances, Hendriks and Yates will be out there, and probably an arm or two that come out of nowhere this season.
At what point does it make sense to tie that money up now rather than pay Giles his arb salary, and delay the investment until you have the maximum amount of information on the health and projection of these relievers going forward? I feel like that point is a number that probably won't interest Giles. If you can get him for 3/27 or something then sure, make it happen.