I don't know if I'm a #menswear gent, but I wear a panama hat in summer.
I've also got a similar, but more casual, hat that's made from woven paper that I got in Japan a few years back. It's crushable and rollable, so you can easily pack it into a bag and then just jam it on to your head and it regains its shape.
If you're not keen on panama hats, then I'm not sure what else you can wear. Flat caps or newsboy caps generally look odd with a suit or odd jacket and trousers and summer's generally too warm for a fur-felt hat.
There's always the trusty bucket hat, but that doesn't look too good when you're dressed up, either...
I have one of the hemp panamas from Akubra (the Bondi from memory) which is ventilated and takes a lot of hard wear and tear.