The common slouch hats worn was called a "slouch hat."
From 1853:
From 1887;
The slouch hat was synonymous to a degree with a soft hat, or unstiffened hat.
They were contrasted with the common men's formal hats of the period, which where stiff as a board.
General Meade in his slouch hat:
civil-war-picket.blogspot.com
General Grant wore a slouch too...
Same with Sherman...
To Contrast a slouch hat, President Lincoln did not wear a slouch hat. he wore a common men's hat of the "fashion" of the time.
Slouch hats, generally black in at least a nod to the army regulations, were widely worn by Union officers.
Sometimes the Army's uniform hat was drawn and worn as a "slouch hat"...
The U.S. ARmy's regulation Uniform hat was stiff, and cocked on one side. But the Union officers and soldiers often wore them as a slouch hat instead...
Or like some officers they purchased citizen's "slouch hats" for army wear...
Confederate officers, like their men, generally wore slouch hats too. From General Lee on down.
The slouch hat was practically uniform in the Confederate army, although it was no part of any official regulation.