When I was a kid, my grandpa showed me a felt badge that was given to him when he returned from WWII. He told me that after the war, soldiers were not supposed to wear their uniforms but that street clothes were in such a short supply that some had no choice. The badge, he said, was supposed to indicate that he was, in effect, not wearing his uniform, even though he was. When he passed, I asked my parents to save that badge for me, but they never found it in his things. It has bummed me out ever since and I have never been able to find out any more info or even see another of these badges. Do you know more?
What you're referring to is known as a "ruptured duck." It was issued in both patch and pin format to signify an honorable discharge from military service. They were sewn over the right breast pocket of the uniform.
I have my grandfather's jacket from when he served in the 9th Air Force, and it has a ruptured duck on it. I'll take a picture tonight if you like.
Edit: Sorry, I got home late and then had family matters to attend to late last night and this morning. I'll post a pic if I can this afternoon.
The ruptured duck, and the jacket.
Here you go.