The peaked cap worn by most civil war soldiers of both sides could be termed a 'shako'. Unlike a Napoleonic shako, however, most were not stiffened to stay upright in the same style of the old French shako. BTW - shako = peak (Hungarian - csákó) The unstiffened 'forage cap' of the French army was the kepi (little cap) very similar to the Civil War caps. and was worn off duty, on exercise or on campaign.
The shako at this time, like the bearskin, was ceremonial or full dress wear and would not be worn in the field. Although there were some units with bearskins in parade dress, they would not be worn on campaign. Most parade headdress in the USA and Europe were replaced by parade helmets from 1870 onwards. The current french kepi is a stiffened 'cap' for formal wear only and replaced the shako in 1870