This is actually a legitimate OSHA complaint. Your employer is required to furnish or allow appropriate PPE, excepting work boots (they don't have to furnish boots unless highly specialized footwear is required, i.e. wouldn't or couldn't be worn outside of your work environment). Clothing protecting one from the elements is required, they can't tell you that you must wear nothing but t-shirts in the dead of winter to MVAs. If you lack natural protection from the sun they either need to supply sunscreen or headwear.
At my 911 job it was Company logo'd ballcap/beanie only.
At my transport job it was plain black baseball cap / beanie or company logo'd fitted ballcap.