A cap with a removable neck cape, for the exposed approach
Outdoor Research Sun Runner Cap
The cape covers the ears and neck that burn first and that no sunscreen ever gets reapplied to; you will look faintly ridiculous, and everyone who has burnt their ears at 4,000 m stops caring.
Skip it ifWinter trekking, where the problem is cold rather than sun.
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The rest of the shelf
Most useful single itemDust mask, neck cover, hat, eye shade
Buff Original Multifunctional Headwear
The best value item in this entire catalogue — it does five jobs badly enough to be worth carrying two, and none of them well enough to replace a real hat.
Skip it ifNothing. Take two.
Best warm hatWorn from four in the afternoon until you fall asleep
Smartwool Merino Cuffed Beanie
Merino stays warm when it is damp with sweat, which a cotton beanie does not — it is thin enough that in real wind you still want a hood over it.
Skip it ifSub-zero summit mornings on its own. Layer it under a hood.