You definitely want to have a cap for Point Lookout, but you are welcome to wear a hat for any time other than formations in which you are required to be in a proper uniform.  
Like many photos of Yankees throughout the war, the New Hampshire men at Point Lookout can be seen wearing the common styles, usually typified by dark colors and modest crown heights and brim diameters.
J.T. Ferrell will be offering two options, the first with a telescoped crown, and the second with a particularly low flat crown and narrow brim.