Yes I am referring to the insignia. If there is a traceable provenance then it would be worth quite a bit but many people add insignia to caps to enhance the value, attributing it to famous regiments without it ever being anywhere near a famous battle or regiment. That battle regiment by the famous big horn battle would have had different smaller insignia. False embroidered insignia is for officers as well so the hat would also not have general service buttons generally speaking. There were no survivors of that battle so the hat would not be in such a great condition but there are other regiments that might be 7th cavalry of some other state.
Just to sperg-out a little; before Custer headed with his main force into what would be his final battle, he dispatched a column under Capt. Benteen down a different valley to check for Indians. Custer's pack train with supplies lagged behind his advance. When Custer came in sight of the Indian village he sent a force under Maj. Reno to begin the attack on the closest end of the village which sprawled for several miles along the river valley while he took a force of around 250 to attack the village from another direction. Reno's force was quickly outnumbered and overpowered causing him to retreat for high ground; a bluff along the river. The Indians kept pressure on his force while most warriors moved to meet the threat posed by the force under Custer himself. As Benteen and the pack train caught up to Reno's position a fight could be heard several miles away. It didn't last very long. Benteen and the pack train remained hunkered down with Reno. The Indians celebrated that night but knowing there would be retaliation they decided to get out of there and didn't seriously re-engage Reno's command. The point of that long-winded description is that about half of Custer's command survived the battle. (Apologies for any inaccuracies, I'm doing this from poor memory.)
You might be tempted to say, "Wow, that kepi might have survived the battle after all!" Not so fast! While most formal portraits of officers of the 7th Cavalry might show them wearing a kepi, that was not the typical hat worn on campaign. The typical hat would have a brim all the way around to protect from sun and rain. There are some additional hat facts that become apparent if you look through the book, "Artifacts of the Battle of Little Bighorn," by Will Hutchison. It contains pictures of items that belonged to participants of the battle from both sides. Not everything pictured was present at the battle but they are personal items that illustrate something about a participant. I recommend the book highly, it has great pictures, and I like pictures.
This first image from page 60 shows Maj. Reno's kepi. On close inspection the button is not general service. It looks more like a staff button.
This next image shows the hat that Capt. Benteen wore on campaign in 1876. It is the officer's version of the Andrews hat, or pattern 1872 campaign hat. It folds flat. If I recall correctly, most soldiers didn't like it.
The next image, on page 68, is the kepi of Capt. Miles Keogh. The crossed sabers and number 7 appear to have been sewn into the fabric and the button on the side is general service.
The next image, on page 94, shows the hat worn by Col. John Gibbon who commanded the 7th Infantry. This was the column that marched from Montana and met up with Gen. Terry's column from Ft. Abraham Lincoln just before they arrived at the aftermath of the battle. He wore this hat while on campaign. Notice it's not a kepi. Gibbon lead the Iron Brigade during the Civil War.
Finally, on page 95, we see the kepi of Pvt. John McGuire. The buttons are general service and the crossed sabers are not false embroidery.