Only if we can turn it off. I let my LPs sit at 0 stamina sometimes because I am getting that last little bit of grog to upgrade them. Luckily, they don't need stamina to train....
I would rather an ability to start it even if they aren't all the way at 0, to top them back off ahead of time.
You would still have to rest them, which would limited how often they could be used, and they would be limited to 10-20 attacks a day at most still. You could just limit yourself to 15 attacks instead of 20, and then rest them earlier, instead of forcing yourself to waste the last couple stamina to put them to bed.
I like the idea @Skye but as @Robin Blood states it does define the purpose of stamina. However . If you would make the algorithms so that starting recovery early would result in a bit longer recovery time.So not 1/20th of the time for 1 stamina but way more at first. And mid way 10 would take 18 hours for example. There must be so one who can make a nice table for that...!