Ok, as someone who's dealing with a slightly bigger problem (Rats) Here's what's actually gonna work (advice from the professional people taking care of my problem right now). (We have a small mouse problem too.)
Secure your feed. Don't leave food out or in the coop. Only feed as much as the chickens need each day, pick up excess at night and keep it all in metal bins when not actively being fed.
Start elimination by determining where the mice are traveling. What walls are they using, where are they being seen. This is where you want your removal methods to be.
Set bait boxes. Bait boxes are secured in ways that the chickens can't get into and are safe to use around them. I do suggest walking the run every day to make sure no mice died where a chicken can get at them to eat them, but 99% of the mice will return to their nests underground before passing away so the risk is minimal.
If you choose to set snap traps make sure they are secure from the chickens first as they can break small bones. Then start by putting the traps down baited but unset for a week, renewing the bait whenever it's taken before you set the snap trigger. Then after the trigger has been set for a week or two after that, move the traps somewhere new and start over again.
If you can find their burrows you can use a powdered poison called tracking powder that they walk through then later lick off of their paws. You just dust the opening tunnels to the nests with it.
Mice can fit through gaps the size of a quarter, sometimes smaller so make sure you're filling in gaps in your walls with 1/2" hardware cloth or just straight up concrete or thick wood. Try to seal every wall so that it's solid. Even mice can get through plywood given a bit of incentive.
Track down areas of debris in which the mice could be nesting and clear them out, whether natural or just clutter in a garage. Lift bales of bedding material off of the floor by 1' on cinderblocks, pallets, etc. opening up air underneath them. Pull plywood or tarps away from walls or make sure they're completely flush and compressed to avoid creating gaps for mouse-only travel that keeps the mice safe.
Cut grass and shrubs low to the ground and clear out the fallen plant matter. Rake your lawn regularly to prevent the buildup of thatch that mice could travel under. Make the area unhospitable and exposed for them.
Don't grow vegetable plants that mice enjoy eating like corn. Only grow vegetables that are mouse resistant like alliums, spices or hot peppers.