I have a long sleeved picnic gingham shirt that I rely on. May be an old Arrow, or some sort of Sears cruisewear, the tag fell off a few years ago. It is tissue thin. Not afraid of those big squares, me.
This shirt here by Penfield (Saxton) nails it as a short sleeve though. Great gingham and throws in a little bone with the workshirt-ish buttoned-up pocket. Found it via a UK stockist but you can buy it direct from Penfield.