take a scarf or sarong with you
I happen to use a shemagh, but sarongs also work great. This simple piece of cotton cloth is one of my most useful travel accessories with many different practical applications.
It’s great for sun protection, a makeshift towel, carrying stuff around, an eye mask, a wrap for conservative countries, and much more. I can’t tell you how many times a scarf has come in handy for me on my travels.