Added another photo.
Vickie, I was hoping somebody here would immediately recognize the emblem of the Fraternal-society kepi-hat. Since nobody has spoken up, I'll do the best I can for you.
Back in September 2011, a digger posted an unidentified crossed-swords insignia. The result was one of the lengthiest relic-ID discussions ever in the What-Is-It forum (over 600 replies to his ID-request post). It turned out to be an insignia from the Fraternal-society "Knights of Pythias, Uniformed Ranks." The Uniformed Ranks was the military-style branch of the KoP. In doing research for that discussion, I came across a great many Fraternal-society insignias. I also saw some KoP-UR fraternal kepi-hats very similar to yours. Most date from the 1880s into the 1930s.
The "embroidered" emblem on your kepi-hat shows three weapons, crossed. One is a short-sword, one is a battle-axe, and the other is indistinct, but resembles a lance. According to my KoP-UR research, members of that military-themed organization "were drilled in the use of the sword, axe, and lance." A KoP-UR coat-pin showing those three crossed weapons is here:
Knights of Pythias
I'm including the photo below. Look at the pin at lower right. It shows a short-sword (sword's handguard at bottom) crossed between two battle-axes.
Instead of having a leather chinstrap, your kepi-hat has a cord as the chinstrap. That is a very unusual feature. (By the way, the chinstrap is supposed to rest atop a kepi-hat's brim, not be stretched across the hat's top.) The cord on yours is white. KoP-UR kepi-hats have a white cord as the chinstrap. The one shown in the other photo below is darkened with age and handling (and so has the emblem-embroidery, which also used to be white).
Some KoP-UR kepi-hats had a wide band above the brim... like yours does. See photo below.
Also, emblems which are made of embroidered-thread on the front of KoP-UR kepi-hats were white. See another photo, below. Your embroidered emblem is white.
Lastly, your hat's manufacturer, the M.C. Lilley Company, is known to have been a big-time supplier of Knights of Pythias uniforms and related regalia (such as hats.) See the photo of an M.C. Lilley sales-catalog of KoP and KoP-UR items, below.
So, for all the above-mentioned reasons, my best "educated guess" is that your kepi-hat is a Knights Of Pythias - Uniformed Ranks hat, and emblem.