The one chinstrap button shown in that discussion's photos is a cuff 3-piece US Army Officer button dating from 1854 to 1902. The inside of the hat has the maker's-mark of M.C. Lilley & Co, which dates it to no earlier than 1872.
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.
_This a is "fatigue cap" according the the M.C. Lilley catalog of 1891. The cap cord is correct over the top.
As stated in previous posts, Lilley was a perhaps the largest supplier of military and societal uniform accoutrements and accessories. This includes musician uniforms. Post Civil War, many communities had uniformed bands and drum corps.
Your hat is in amazingly good condition. If it were 100 years old, I would expect the red grosgrain to be faded. Whatever the age, it never saw much use.
Thank you everyone, especially CannonballGuy and Bramblefind for always helping. This hat belongs to a friend that can't leave his house because his wife is very sick and I'm just trying to help him out, but couldn't without you guys. Thank you very much.