The US Army regulations of 1858 called for a forage cap of blue cloth. These were mass produced for the enlisted soldiers. There were some minor variations among those produced by contractors to supplement the military production at the Schuylkill Arsenal facilities at Philadelphia. They were of a simple pattern, without a separate band, and no particular stiffening inside other than at the top of the crown.
The officers had to purchase their own uniforms from tailors or outfitters. There were variations on the "shape" of their forage caps based upon their taste and the company they bought them from.
From the Uniform regulations...
For example, General McDowell and many of his staff wore variants with a crescent shaped visor, and rather taller crown than found on the Government issue forage caps.
Here's an example purchased/worn by the chaplain of the 6th Vermont Volunteers.
Others, for example, like General McClellan, preferred for a forage cap something slightly more akin the French "kepi" pattern, with a separate band (somewhat stiffened) and rather lower crown, and straight visor.
Here's General McClellan's forage cap:
Many Union soldiers, disliking the somewhat shapeless Army issue forage cap, privately purchased blue kepi's the soldiers called "McClellan caps."
Other's referred to them simply as "kepis" or "chasseur caps." But McClellan cap seems to have been popular with the Union soldiers. A veteran of the 21st Massachusetts Volunteers recalled..
Here's an army medical steward wearing a privately purchased "McClellan cap" as a forage cap...
Among the general issue forage caps provided to the troops by the Government, as mentioned there were slight variations based on the maker. Mainly in the width of the circular crown piece, and perhaps shape of the visor. Here's just a sample of some late war forage cap producers under government contract...
Irrespective of the maker, once delivered and paid for, these forage caps were distributed to the troops indiscriminately despite any minor variations in pattern, color, etc.; just "forage caps."
In recent decades, Modern collectors etc. classify the US Army government issue forage caps as "type I", for example with a narrower crownpiece, and "type II" with a wider one, for their own purposes.
Similarly, it has recently become customary to refer to the forage caps with a crescent visor and tall crown, as a "McDowell" pattern cap. Though the term was unknown in the 1860s.
The kepi/Chasseur/McClellan caps were common among the various Militia corps too, which wore commercially produced clothing from outfitters too. Like these fellows of the 22nd New York State Militia in 1862.
Or the 7th New York with their Gray ones...
The 7th New York State Militia had adopted these "fatigue caps" (as they called them) in 1859, to be patterned after the French style we often call the "kepi."