WWII German SS Peak Hat

I am having trouble finding a hat that is exactly the same as this one. I found a replica hat for a Waffen SS Storm Artillery but the hat is quite a bit taller than this one. Under the internal leather band, I found a white, loose-woven patch with possibly some kind of symbol and writing stamped on it but it is so badly faded I can't identify them. Someone has also written with ink some numbers and possibly letters under this band. A patch of clear plastic material has been sewn onto the hat lining where it would rest on top of the head.. It is not sewn on one side so as to create something like a pocket. On replica hats I have found in photos, the inside of the peak of the hat does not have the same chequered pattern embossed in them as this one. The hat is dark green and appears to be made of wool. It has red piping. The lining is a sort of gold satin. The black band to which the skull and crossbones is attached is a velvet material. In a couple of places inside the hat, I can see woven timber which goes around the entire hat and makes it stiff. It smells like a very old piano! :) It feels very fragile, has been hand sewn by someone in the past but appears to be very old. Do you think it is genuine? I am very interested to confirm which part of the SS this hat belongs.

Poor