This is the period when the US Army adopted the "Montana peak" style, officially on September 8, 1911. As to their origin one can only guess. Previously in the Spanish-American war and Philippine Insurrection many troops peaked their M1883, M1903 hats this way rather than the regulation fore and aft crease, in order to shed water better. This service hat has 4 ventilation holes on each of the peaks, all are perpendicular, to the 3 inch brim.
Hats of this era have the following characteristics: 2 to 8 rows of stitching around the edge of the brim, most common is 5 rows, no grommets for a leather chin strap on the brim, and a high peak of about 5 3/4 inches in height, (measurements are taken from the inside of the hat). Also underneath the sweatband are two small holes with circular brass grommets. This is to let the shoestring chinstrap pass through and underneath the sweatband down to the chin. The sweatband itself is not sewn to the body at this point in order to let the strap pass through. The shoestring tie runs underneath the grosgrain ribbon behind the ears and at the back of the hat. A makers trademark or name may be present on the sweatband, usually this applies to General officers' hats. Most of the ones that were enlisted personnel did not.