If Bierce had written this story in causal order, he would have begun with the background on Peyton Farquhar. Then he would have noted Farquhar's discussion with the Federal scout. He'd then skip ahead to Farquhar's execution. His dream occurred between the time he was in the noose until the time his neck snapped when the rope ran out of slack. To describe his hallucination/dream in a linear fashion, Bierce (or the narrator) would have had to write that in somewhere. As the board was thrown from under Farquhar's feet, his lavish hallucination would be condensed to, in actuality, a second. The narrator might have written something like, "During those fractions of a second, Farquhar envisioned his escape, a romanticized tale of heroism proving that maybe all is fair in love and war. However, at the fantasy's climax, the rope snapped him back into the reality of his death."
That being said, Bierce could have chosen to wait until the end to reveal that this was a hallucination; he could have done this with the story in linear order or out of order. But writing the story out of order does play with the idea of time. Farquhar's hallucination, which seems to take place over many hours, actually occurs over a matter of seconds or minutes.
In keeping the elements out of temporal order, the narrator gives the reader a sense of things being out of time; the reader can more easily become immersed in Farquhar's fantasy.
Prior to the ending, the reader could interpret certain sentences to indicate that Farquhar's escape is in his mind. One of these instances occurs at the end of Part I:
As these thoughts, which have here to be set down in words, were flashed into the doomed man's brain rather than evolved from it, the captain nodded to the sergeant. The sergeant stepped aside.
This does seem like a clue that "these thoughts" indicate Farquhar's escape is (will be) in his mind. But this is interrupted by Part II.
Perhaps most importantly, by placing the background section (Part II) in between the moment the sergeant gives the order and the moment Farquhar falls to be hanged, a temporal space is created. The reader literally has to wait, the time it takes to read Part II, for a duration between the order to hang him and that order to be carried out. This gives the reader a real sense of time between the order and the hanging. This is the stretched out time of Farquhar's hallucination. A flashback or a vision (hallucination/dream) can be detailed and elaborate but in a narrative, each can be condensed to a few moments in "real time." The non-linear structure helps establish this idea.