The Clever Disguise was worn by John in an effort to avoid detection by his Dad. It was originally composed of the Magician's Hat and the Beagle Puss. John changes the disguise twice before encountering his father.
Despite the rather distinguished upgrades, Dad still manages to seethe disguise.
The Clever Disguise has a built-in defense, called the Beagle Aegis. This appears to diminish loss from the Prankster's Gambit when John was hit by Dad's pie; however, judging by John's heavy loss of Gambit, it was not very effective.
At some point in his time travels, Davethe Clever Disguise. Jane Crocker is seen with a similar disguise, also wanting to sneak out and get mail without Dad noticing her.