The Boss of the plains hat didn’t have a high crown. The crown averaged about 4 inch crown and a 4 inch brim. The flat brim shielded the cowboy from the sun and weather. The cowboy didn’t shape the crown or brim. The brim would get bent in the strong wind and when they were galloping their horse. The Boss of the plains hat would get creases from the cowboy when he would grab it by the crown.
As seen in this photo the crowns are 4 inch and the brims are 4 inch or less. The Boss of the Plains was the first cowboy hat.