You can get 'sun & swim' protection for the hair that creates a barrier to chemicals.
Its usually just a spray-on, comb-thru thing.
We in the UK have this, but I'm sure there are equivalents in other countries.
http://www.boots.com/en/Boots-Sun-Swim-a...l_1262097/

The hair will look perfectly natural when you're swimming.
When you emerge from the water, I usually come straight up out of the water and give my head a sideways flick.
That way all the hair is hanging down, with the front swept to one side.
You'll then look like anyone else with soaked hair.

You just need to make sure your attachment is recent & solid on the day you go swimming.
You'll be fine!

PS. You could always emerge from the water having had this mishap and you'll find no-one is looking at your hair: