I tend to use decathlon ski socks. I like long socks so I can have them pulled up when I want extra warmth or scrunched down if I am too warm. I do the same when running.
I tend to wear Bridgedale Trekker or Summit (I think the names have changed now) but they are over your budget. I've recently been trying out some socks from the Decathlon hunting range which have been really good so far, very comfortable but I haven't tried them in anything really cold yet. They do say they are designed for "biting cold". They are about £8 or £9 a pair so maybe worth a shot
> I usually go for two pairs: ski-socks over thin running socks.
Strangely, I stopped getting blisters when I stopped doing this. I've heard a few others say this too. Your mileage obviously varies, but it can be worth trying solo.
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Podcast Mountain Air - 14. Dougie Baird, Mountain Path Builder