Called a Hardee hat.
Later replaced by the "Witch hat" a black campaign had that allowed the brim to be hooked to the crown. Poorely made since if caught in the rain you could wind up wearing the brim like a collar! Then replaced by another black hat which is correct for the Indian War period, and much much later by the tan campaign hat which hollywierd seemed to equip everyone with but didn't come in until the end of that period.
Then I always wondered why there were no privates in hollywood movies since everyone was wearing NCO trouser stripes!
Yellow suspenders? Bull pucky, pants were tailored to fit and even when suspenders came in very late in the period, they sure as heck were NOT yellow!!!!! Same goes for the yellow neckerchiefs. Any colour was worn, depending on what the sutler had in stock.
Now did you ever wonder why the movies show guard mountings in campaign dress, when full dress used tunics, and Spiked or Plumed helmets resembling the Prussian style? And that was true even on the "frontier." In fact an inspection report in the llate 1870's wrote up a couple of soldiers for not having regimental numbers on their spiked helmets!
Last of all, rank insignia was NOT worn on campaign shirts. The army was small enough that you damm well knew who your NCO's were!
Add to that brass insignis was not worn in the field on campaign hats. Why provide a shiney target for the apaches, who unlike the plains Indians were very good marksmen.
One last bit. Garrison dress was a five button blouse with rank chevrons and kepi, NOT campaign dress.
Still more. Stable dress was brown canvas tunic and pants. When the Rough Riders, were formed there was not enough blue uniforms to go round so they were issued stable dress. Something which proved far more useful than BLUE WOOL in Cuba, and a lot cooler.