The history of the cowboy hat as a staple of gambling culture is not documented, but if we were to speculate, gambling originated in the Wild West, where cowboys and townspeople all wore such head coverings.
Cowboys would use their hats indoors to cover their faces so that their opponents would not be able to read any tells off during a saloon game of poker, for example. Then, popular culture came along, and the gambler-style cowboy hat was taken at face value.
InClint Black’s song “A Good Run of Bad Luck,” he is wearing a cowboy hat and singing about gambling, for example. There are other cultural references just like this one!