Hi! When you regain energy, that restores your normal walking speed until you run out of stamina again. However the exhaustion effect still applies (unless you use
muscle remedy), which should be indicated by the tired icon above the energy bar:
In my tests, going to sleep exhausted did correctly reduce the stamina recovery.
However, I noticed one issue while looking into it: sleeping late while exhausted is meant to
reduce your stamina overnight, but that didn't work correctly. That's fixed in the upcoming Stardew Valley 1.6.9; thanks for reporting it!