What's with the hat? SPOILER

Deborah Kerr transports the dead spy in the wheelchair across town with his hat pulled down over his face. This is to fool townsfolk into thinking it's the old man that she takes care of. But when she gets back to her romm, she spots a hat on a table and panics, the implication being that it belongs to the spy. But had his hat on! To top it off, during a nightmare she has concerning the body, he clearly is wearing his hat. When she wakes up, the hat falls out of the chimney where she hid it. So whose hat is it?