When the POWs are running from the barracks to line up for the appel, the Chief is not one of the men running, yet in the next cut, as the men are lined up, he is in the front row of prisoners.
Max tells Larabee to call him at the airport and make him sound important. Moments later, he is paged as "President Maxwell Smart". However, when he answers, it is a call from 99, who would not have known to make him sound important.
The Chief tells Max that the fence around the camp is electrified constantly, however, when Max crashed through the window of Siegfried's office and was cornered by the "dogs", he could clearly be seen pressed up against the fence with his hands holding on to it.
Siegfried tells Kessler they should go inside because it is too hot, however, both he and Starker are wearing heavy leather coats at the time.
The arrival of Major Kessler is treated as a big deal, and Siegfied treats Max (who he initially thinks is Kessler) as a superior. Bit Siegfried is referred to as Colonel Siegfried, so he would outrank Kessler by two grades.
When Siegfried is addressing the prisoners, you can see vapors forming from his breath - it's clearly quite chilly. (This is especially noticeable when he says, "The picnic is over" and "What is it, Chief of Control".) But just moments later, he remarks that "it's much too hot out here".
There are two kennel doors visible below the window of Siegfried's office. However, it is obvious they are a faux facade: Doors that tall would require the floor of Siegfried's office, and by extension the window, to be several feet higher.