No one actually makes improvements from training. Instead, they make improvements from recovering from training. Rest and sleep are essential, but too many people take the concept of recovery to an extreme. Unless you ran an ultra-marathon while carrying an Atlas stone, you don't need to spend your days off emulating a corpse on the sofa.
Instead, practice active recovery techniques like Prowler pushes, kettlebell swings, sledgehammer work, med ball throws and slams, or even riding a bicycle up some hills.
Weight training involves both eccentric and concentric movements, but it's the eccentric (lowering or negative) movements that can cause damage to the muscle and increase soreness, and it's essentially the eccentric activity that you need to recover from.
The best active recovery exercises are non-eccentric or mostly non-eccentric movements. As such, you can do them on your recovery days to increase overall blood flow, which improves recovery, and burn a few extra calories after a "rest day cheat meal" that would've otherwise been stored as fat, to further your physique-honing progress.
You don't have to do these movements on every single off day from the gym, but there's no reason you can't do them twice a week and have one day off for complete rest. For instance:
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