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hats-
Look at Outdoor Research. You can't beat them for quality, breathability and weight. They are beloved by mountaineers.
Jana has this one. It rolls up and packs/stows well and has loads of ventilation. She wears it in the mountains in summer when she's not wearing a helmet: https://www.outdoorresearch.com/us/sombriolet-sun-hat-243441
They have lots of variations though: https://www.outdoorresearch.com/us/womens-sun-hats
I usually use a cap with neck attachment but on really hot days wear a US army surplus boonie with neck cover. Much cheaper than the OR but much heavier
Stonehedge
Not saying they're any more likely to suit you than a bucket hat but can highly recommend US/AUS military Boonie hats for keeping sun off all day.
I look like an utter plonker while wearing one but I haven't found anything better for keeping the sun off during a full day in the mountains on a high UV day.