birdfeeder attracting mice

I've had a bird feeder for about a month now, and at first it was only attracting birds and squirrels. I didn't mind the squirrels eating from the feeder as I was more than happy to refill the feeder every day. But recently I've noticed mice eating from underneath the feeder too, and I can't seem to get rid of them. I've tried cleaning up under the feeder (on the ground) everything I can, though this is difficult because there's muddy ground there, as my feeder hangs from a tree (there's nowhere else for me to put it)
I've tried spraying peppermint oil which I heard deters mice, but it doesn't seem to be affecting them at all. I really don't want to stop feeding the birds, as it's almost winter and I don't want to cut off their supply, and I also have a lot of unused bird food which I wouldn't know what to do with, if I were to stop.
Does anyone know how I can keep the mice out of my garden, without stopping feeding the birds? I don't really want to put down rat poison because the birds could eat that too, and there's nowhere I can put it where the mice will eat it and the birds won't
I've tried spraying peppermint oil which I heard deters mice, but it doesn't seem to be affecting them at all. I really don't want to stop feeding the birds, as it's almost winter and I don't want to cut off their supply, and I also have a lot of unused bird food which I wouldn't know what to do with, if I were to stop.
Does anyone know how I can keep the mice out of my garden, without stopping feeding the birds? I don't really want to put down rat poison because the birds could eat that too, and there's nowhere I can put it where the mice will eat it and the birds won't
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Bird food is like a banquet for mice and rats. Fortunately I live in the country and both grow and am surrounded by plants heavily laden with fruit, seeds, cobnuts etc etc. I also don't use insecticides so there are plenty of insects. All in all there is more than enough to sustain the birds. I don't want to attract vermin to the garden so I have, therefore, made the decision not to put out additional food for the birds except when conditions are particularly harsh and the other supplies have finished (February usually).
Where there's bird food there will almost certainly be mice and possibly rats. Mice I don't mind - rats I do.
I do, however, always put out water for the birds - unless there are lots of puddles
@wild edges's advice is good re trays under the feeders. The mice here use our feeders in situ as they're against the fence and they can access them from there, where they hide.
I often saw him/her feeding with the birds too - I have a pic of a blackbird, a dunnock and mousie all out together on some paving during last winter
What a great photo too.
I havent anything that good. My camera isn't as good for close ups.
Taken through a window though.
Sadly no longer with me