I believe that it actually did exsit at this time as I'm pretty sure I've seen era pictures and a book showing era hats, weapons, uniforms, equipment, etc. with it in it. Longstreet's hat is either a Hardee hat or a Slouch hat. Both fairly common among officers on both sides.
They did have that kind of hat back then, it's a simple slouch hat of any kind of generic quality. If you want to talk about hats, well, the hats that people like the Chamberlains and other actors turned re-enactors had were, for the most part, cheap and inaccurate reproductions. The chin straps are too thick and a lesser quality leather, the bill of the hats are cut wrong, the wool is wrong, the dye is wrong, the only thing right is the buttons.
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