In the sequence where Schultz interrupts Klink after waiting two hours, Hogan comes in as Schultz is leaving. In all the shots from Klink's right side (showing Klink and the office door), there is a mirror on the wall above Klink's left shoulder, right beside the window curtains. When the view changes to directly in front of Klink, and he stands up to talk with Hogan (standing at the left end of the desk from Klink's point-of-view), there is no mirror on the wall, only a picture with two images - the top image appears to be a submarine.
Col. Klink is not married. Yet, early in the show, Col. Klink is wearing a cameo-style ring on his left-hand's ring finger. Later, it appears that he is wearing a traditional wedding ring, or he has a band of "lighter skin" showing (no ring). Even later, he is wearing the cameo-style ring once again.
In a conversation with Klink, Hogan mentions having been assigned to duties "...at the Pentagon..." However, the dating of events in the series indicate that Hogan was an inmate of Stalag 13 from February of 1942 (the date given for events in the pilot) through the end of the war. The Pentagon was not dedicated until 15 January 1943 (but was partially occupied as of 1 April 1942). Since Hogan was already in Stalag 13 by February of 1942, he could only have been employed at the Pentagon as a construction worker, not an officer.
When Carter tells Hogan to guess what the rabbit trap is, Hogan guesses, "It's bigger than a bread box?"... a line coined on the 1953 quiz show, "What's My Line?"