Geoff Leighton is a middle-aged man in England who has just gotten back to his job as a conservationist at a museum after taking time off while his mother was in hospice. His boss shows him a new donation for him to restore. It is a large chest. Inside, Leighton finds a robe, slippers, a dagger, and other parts of a magician's ritual kit. He also finds a letter saying that the objects are from Dr. Marmaduke Palmes, who was a surgeon in York. The collector writes in the letter that he will have none of it, and cites a bible verse. Below the magician materials, Leighton finds a coat made of patchwork pieces of leather. He finds a note in the pocket reading "John Palmer: his Innocent Coat." 
Leighton takes a break and goes to lunch. When he returns, the coat is hung over the back of his chair. He is angry because that is bad for the fabric, but his only other coworker who is there says she did not touch it. Leighton goes home for the day and gets rid of his mother's boxes that are in his house. That night, Leighton has two dreams. In one, two men are meeting in the corner of a pub. The older man hands the younger one a package, and says that he will go blameless. The younger man gives the older one, who is his uncle, a bag of coins. The uncle warns him to wear whatever is in the bundle against his own skin and not take it off. He also warns that there is a price to pay. In the second dream, Leighton is at the top of the stairs, sensing and dreading someone coming up. He then sees someone, lying flat on the stairs and crawling upwards.
The next day when Leighton goes to work, the Innocent Coat is missing. He asks his coworker if she knows what an Innocent Coat is, and she does not, but she does know who John Palmer is. John Palmer is a false name used by a highway man in the early eighteenth century named Dick Turpin who was hanged for a double murder. Leighton tells his boss about the missing coat, and then begins to do research on Turpin. He is perplexed by Turpin's behavior and how he acted as though he was going to get away with everything right up to the end, when he stepped off the gallows voluntarily but took five whole minutes to die. He also discovers that a few days after Turpin's death, his grave was robbed and his body was discovered at the house of Dr. Marmaduke Palmes. The body was reburied naked and Dr. Palmes did not get in trouble for the grave robbery. Leighton reads a physical description of Turpin, and he looks like the younger man in Leighton's dream. This makes Leighton feel unwell, and he leaves to go home.
On the way home as he is waiting at a light, Leighton hears two men arguing behind him even though there is nobody there. One man asks why the other did not wear it always like he instructed, and the other answers that it gave him evil dreams. The first man, who is the uncle and Dr. Palmes, asks where it is, but the other man, whom Dr. Palmes calls Richard, says that the voices told him to trust them and that the Innocent Coat will save him even at the last drop. Leighton smells leather, but he turns around and nobody is there except an old woman. He goes home, and discovers the Innocent Coat on the floor of his home. He does not want to bring it back to the museum, because his boss will think he stole it. So, he decides to get rid of it.
Leighton remembers the verse that was written on the note in the chest and decides to look it up. The verse is about a "fretting leprosy" of a garment that should be burned. So, he decides to burn the coat. He puts it in the fireplace and leaves to get matches, but when he has returned the coat has disappeared. Then, Leighton gets a phone call from a witch museum that he had previously contacted to ask about Innocent Coats.         The man on the phone informs him that anyone who wore an Innocent Coat could get away with anything without suspicion or the law coming after them. The coats are made from the skins of hanged men that were murderers. 
Leighton hangs up and thinks about how the coat is waiting for him somewhere in the house. He thinks about how the jacket would touch him, and remembers how his mother used to hold her hand over his mouth and nose until he passed out. He recalls when his mother was in hospice, and he had suffocated her and killed her, but he could not stand the feeling of her skin. Leighton decides that he cannot stand the thought of the coat's skin pressing against his, so he takes a potato peeler out of his drawer and starts removing his skin.