I think I know why you have struggled to get a straight forward answer to this. There isn't one.
Big companies = fall. Smaller companies = later. Though that is no hard and fast rule. You will be counting out a lot of the companies that you typically see in the fall (the big list companies), and maybe even all of those that look to recruit from summer interns and top-up their staff.
It depends on how you want to do things. Would you like to be away in the fall knowing you don't have a job and have to return to desperately recruit? Or be sure you have a job before you go away (I am pretty certain you could back out if needs be on a spring study abroad). Or do you not want to recruit for MBA standard stuff anyway, at which point it is mostly individual work anyway?
What I do know is that people who don't have jobs January of their year of graduation look really scared and worried. I am sure they sort it out, but for the most part that seems to be a signal.