Similar to Aziriphale, I find my hair sometimes looks a bit better after it's been flattened by my hat. Which is a darn good thing. Where I live, there are not many days when it's sensible to go outdoors for more than ten minutes without a hat between mid-December and mid-March.
Try a merino wool hat -- the material helps. I don't really get too much of a static problem; I guess that is humidity dependent.
Do you have a puffer with a hood? You could also wear your hood instead of a hat; where you live, you probably don't need both (unlike me...today it was beanie PLUS down hood...wheeee!!!!)
When I run, I use an ear band -- that's enough for not-super-cold temps for me and does not mush the hair up.
Photos of different kinds of toques and beanies for your enjoyment.
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