A "hat stretcher" will work for between 1/8 to 1/4 inch difference.
Pretty much any thing over than, runs the risk of popping the rear seam.
Being woolen fabric with the leather sweatband, forage caps can have a tendency to shrink due to sun and rain exposure. Any round shaped object of similar cap size can be used to stretch, but as shared a "hat stretcher" with its controlled expanding scr*w system gives one greater "control."
In gleichem Schritt und Tritt, Curt Schmidt
Not a real Civil War reenactor, I only portray one on boards and fora.
I do not portray a Civil War soldier, I merely interpret one.