If you give it out with the same calendar frequency, then making exhaustion come back on a short (overnight) rest is a no-change.I'm not against this, but I am curious why. The only two points of attrition that have the potential to last longer than a single long rest currently in the 5e rules are exhaustion and spent HD.
With an exploration heavy campaign, exhaustion is something that can come into play as a penalty for player decisions or for failure from "skill challenge" type of scenes. Say a bad survival role for choosing a camp and dealing with a flash flood and needing save selves, mounts, gears, wagons.
Having a level of exhaustion cured with a short rest takes all the sting out of it unless I'm giving it out with more frequency, or giving out multiple levels based on the severity of failure. The latter I'm not against if there's a good reason to clear it up quicker. Actually, I probably like it better now that I'm examining the idea.
But what brought it up the suggestion?
If you give it out with the same encounter frequency, then making exhaustion come back on a long (week) rest is a no-change.
Ie, imagine non-gritty rests. You are force marching for a day. Do you get 1 level of exhaustion? Well this has no effect if you still get a night's rest.
Now imagine gritty rests. You stay up overnight. Does this result in a level of exhaustion? If so, then staying up overnight becomes ridiculously more expensive; in standard, staying up overnight for a night, then sleeping, then repeating this cycle every 2 days is stable. In this new system, that same narrative set of actions results in death after a few weeks.
My personal system -- 3/8 chance each spent d8 HD comes back overnight, 1 HD can be burned to recover exhaustion overnight -- is somewhere in the middle at low levels, and approaches the "more generous" one at higher levels.