John's clever disguise.
Jane's clever disguise
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.
He initially adds a pipe from the Living Room and, upon entering Dad's study, adds the Bowler Hat. This final upgrade makes the disguise "somewhat less funny, but A LOT more distinguished looking."
Despite the rather distinguished upgrades, Dad still manages to see directly through the 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, Dave wore the Clever Disguise.
The Clever Disguise's role ends after being forced out of the Sylladex and presumably picked up by Dad or Shale Imps after John enters The Medium.
Later, Jane Crocker is seen with a similar disguise, also wanting to sneak out and get mail without Dad noticing her.