Well in the event you decide to dabble in rain coats I’ve had a lot of success with the ones from Uniqlo. Very water proof, insulating layers, not too large, pretty stylish, and most come with hoods that have a degree of structure and a small brim built in so you aren’t getting rain in your face. Also as someone else said r/malefashionadvice will probably give you a lot more insightful answers

If you do decide on/find a good hat though report back.