keeping a large button-up shirt in my bag at all times

I hate summer. I hate heat. I hate the sun. And yet I live in Los Angeles, and I don't have a car, so I am constantly walking around in the heat and the sun and absolutely fryyying. So I keep a button-up in my bag at all times, and I throw it on when I'm walking around. The button-up does not have to be cute, but it should be light and thin and large enough to wear loose. I got most of mine from the menswear section at Goodwill.