Personal preferred headgear for any outdoor event is hat with full brim. No need to sunburn the ears or anything else, unnecessarily.

Long-sleeved "safari" shirts in British Tan, khaki and off white go well with the kilt and do have something odf a "near-issue" martial-ish look.

(1) Tilley (with elliptical 3"+ brim and ventilated ring around top), (2) Equator 3" round brim and "Foreign Legion" optional drop-flap in back and (3) pressed fur felt Aussie Army bush hat cover most non-winter weather. All have a deployable chin-strap, should same be needed.

NEVER a "baseball cap." (wrong face shape for the style)