An umbrella sounds a great idea..why not treat yourself to a pale coloured pretty one that would look nice on a summers day.
I lived in Aus were most people wear hats in the sun..i would really recommend a good one....it also stops your hair from drying out.
I wear alot of linen in the summer.Why not get a few pairs of linen trousers, they are really comfortable and cool to wear and look good too.I think (but check this out) if you get really close weave fabric they will block out most of the suns rays, though not all