You can go the old-fashioned gangster route with pinstripes, a long watch chain and fedora.
Although mostly associated with gangsters, the fedora was by far the most popular hat in 1920's men's wear.
Everyone wore a fedora, not just Al Capone and company.
Fedora Hat with Signature Web Band and Gucci Script Ornament is a natural straw hat with coral and salmon web and gold hardware.
It is a softer straw hat - similar in shape to a fedora - and if it is well made, can hold water and be rolled up and packed without losing shape.