Come and Go Mad by Fredric Brown (first published in Thrilling Wonder Stories, 1949)

• It’s set in a mental hospital, where the narrator/patient tells how he learned a simple occult technique for exchanging minds.
• The “ritual” requires a burnt offering (even a cigarette will do) and saying the exact words.
• While recounting the story, he asks the attendant/doctor for a cigarette; by the end of the tale he has already performed the swap, now speaking from the corridor as the staff member.
• He tells the man now trapped in his old body not to bother shouting because no one will believe him, reminds him about the necessity of the burnt offering and the words, takes the car keys, and walks out.

It’s been reprinted in collections such as The Best of Fredric Brown and From These Ashes: The Complete Short SF of Fredric Brown, and has been adapted for audio (e.g., The Drabblecast).