People originally domesticated cats to control rats & mice.
SO yes get a cat, lock it in the shed overnight and only feed it in the morning.
OR
get a couple of carpet pythons, same story, leave them in the shed and make a nice sunny place for them to lay & digest your problem.
A big one can knock off a dozen or so mice in a single night but of course will not hunt for another week after that.
Snakes have a double effect because they leave a scent trail that mice will not cross unless in a frienzied plage
Field mice are hard to control as they will not take most house mouse baits and are too light to trip most mouse traps.