Ellery Queen walks around the crime scene looking for clues. He is not wearing his hat. He exits the living room and once he enters the other room, he is wearing his hat.
In the library, the call numbers for books on astrology and astronomy match neither the Dewey Decimal System nor the Library of Congress Classification system. Astrology has a Dewey Decimal number of 133.5 (part of the 130s - parapsychology and occultism) and Library of Congress numbers of BF1745-1779, while astronomy spans 520-529 in the Dewey system and the entirety of the QB numbers in the Library of Congress system.