I never really used anything for my face on my route.. mostly just wore a toboggan. As long as it doesn’t hinder your vision (especially peripheral) I would wear what ever is comfortable for you and keeps you warm. Something like this should work. Amazon - Head Gear
Hey!! I finally gave in and got this one for the worst days. Used it today for the first time, worked awesomely. I have a furry headband for when it's not too windy. I need to insulate my whole self in order to survive lmao
My partner has this hat: https://www.amazon.com/Mysuntown-Trooper-Hunting-Ushanka-Windproof/dp/B015E0E3AI/
I can't guarantee that it's 100% vegan since I didn't buy it myself, but it's all polyester as far as I can tell.