Hate to say it, and I'll probably get flamed by some people, but I'm not a fan of barn cats either. I've kept cats for years and I currently have two indoor cats, but no more barn cats for me. They kill way to many birds and other critters I don't want killed, they roam around and get killed by coyotes or fight with other roaming cats and get abscesses and other wounds, one of my cats came down with FIV, it's just one thing after the other with outdoor cats. They also attract stray and feral cats who then spray all over the place. Nope, not for me. And I still had mice!
In years of rural living I have on two occasions resorted to poisoning mice to get on top of a population explosion. Worked like a charm both times but I really prefer to avoid that. These days I keep a few of those electric Rat Zapper traps set all the time. Certain times of the year I catch mice every night, haven't had any for a while now. I like them because they are a quick, humane kill and easy to dispose of the dead rodent. I also like the bucket traps.