I have to go along with the dog idea.
Hay rats and field mice have always been a big problem at our farm. We used to use Purina Assault poison, which worked better than any other poison we used. But still, it only made a significant dent.
Our neighbors have chihuahas and we watched one "doggedly" (pardon the pun) try to dig something out of a heavy piece of metal in our pasture. We finally turned it over for her and sure enough there was a big hay rat straddling two pieces of metal underneath trying to keep out of the dog's way. That dog grabbed the rat like grease lightning, end of rat. So I would think any type of small, superactive dog, like chihuahas or rat terriers would be a great asset.
We don't see the rats and mice at all anymore, but we know at least a few mice are still around because we still do see some mouse droppings but nothing like we used to. The only change has been the addition of our two English Shepherd dogs. The funny thing is they only go to the farm with us on weekends. I've never seen them catch a mouse or rat, but I'm pretty sure they do from time to time when we are inside or leave them alone. I suspect the smell of my intact male spraying here and there has caused them to pack their bags and leave too. Whatever the change is, I am grateful.
Just some ideas....