Certain stereotypes of the era come to mind, Charlie Chaplin wore his legendary small bowler hat. Although I doubt this sparked a trend in men.
As with previous decades men wore hats depending on their class, upper class men wore top hats or a homburg hat. Although things were changing, with the upper classes owning and driving cars they needed a driving hat, which was a soft cloth cap usually worn by the lower classes. Middle class men would usually wear a trilby hat and working class men would wear a flat cap or no hat at all.