How might the impact of this story be different if Bierce revealed the events in chronological order?
By presenting the story in the order he does, and by changing the perspective of the narration, Bierce skillfully draws readers along so that he is able to trick them into believing that the action of Part III is real, rather than imagined. Using Part II as a flashback interrupts the present time scene of the preparation of the hanging. It allows readers to understand Peyton Farquhar's motivations as well as the Union Army's motivations for hanging him without getting bogged down in information that isn't the point of the story. Starting with the hanging scene immediately draws readers into the drama. It creates emotional involvement as well as curiosity, making the reader want to keep reading. Breaking away from the intense scene to a calmer background story gives readers time to decide how they feel about this hanging, and many may come away rooting for Farquhar because he was the target of a "sting" operation, having been deceived into trying to destroy the bridge by the "Federal scout."
Now that Bierce has enticed readers emotionally and further cemented their commitment to the main character, he can maintain their buy-in through the next section, despite the implausibilities it presents. The objective narration of the second section seems to carry over to the third, and readers assume the detached third person point of view is continuing, when in fact readers are experiencing Farquhar's thoughts rather than reality.
It is possible the same effect could have been achieved by changes in narrative point of view only, even if the story had been related sequentially. However, the flashback puts readers off balance, making the trick of narration more difficult to detect. The story would have been less emotionally engaging and the surprise ending would have been less effective if the story had been written in chronological order.
Bierce shifts his narration throughout the story as well as presenting the story out of order. His narration shifts from a conventional third person point of view to a more revealing third person point of view in the second section. We learn more about Farquar's feelings and his perceptions as well. The third section then turns into a first person point of view and we are able to piece the story together. Bierce does this to disorient the reader. He wants the reader to hallucinate along with him and he wants the reader to be as disoriented as he is. This helps to push the themes of time, deception and death and dying.