Factual error: When Mrs. Logan escapes, Aaron Pierce refers to her as "the First Lady" over his radio. The Secret Service has code names for each of the members of the president's family. He should have used her code name when communicating to other Secret Service agents about her.
Plot hole: Martha Logan told Evelyn about the transcript and hid it in her bra in front of her. Yet when Mrs. Logan later tells her husband about it Evelyn doesn't seem to have any idea what she is talking about. While it can be considered a character choice for Evelyn not to speak up, there is no reason at all why Martha, who is so desperate for her husband to believe her, doesn't ask Evelyn to verify it.
Continuity mistake: Season five, 5th hour. Lynn knows about the four people who knew Jack was still alive, he mentions their names and talks to Jack about it. Yet when they are told Tony wants to speak with Jack, Lynn looks completely puzzled and Jack tells him that he knew Tony. Minutes earlier Lynn already knew that.