If we're talking about the light Eisenhower type jacket that was thought to be Oswald's, there's some info out there that shows the label of the jacket had a dry cleaning tag attached to it and the cleaner was located in CA. The story goes that why would Oswald have a jacket on with a CA dry cleaning tag on it? I don't have links to that info but you can look it up.
This is similar to multiple wallets supposedly belonging to Oswald, the shells of an auto gun when they should have been revolver shells and other bullspombleprofglidnoctobuns to frame him up.