Multiple comments on this question post about "The Woman Taken in Adultery" pericope are wrong, based on the text-critical issue (along with narrative and redaction criticism issues) of John 7:53-8:11 most certainly not being part of the "original" Gospel of John. I put "original" in scare quotes because the original version of John did not have John 21, most likely did not have John 1:1-18, and due to various Gnosticizing vs. anti-Gnosticizing battles, did not have John 6 in its current form. Reminder: a few manuscripts place it in the Gospel of Luke, and in two different places, no less. The link also notes what critical scholars know otherwise: it grossly interrupts the flow of the Johannine narrative. (Bible Gateway, the passage link, also notes that a few mss put the pericope after John 7:36, a worse interruption of the flow, or at the end of the entire gospel, which, as with the mss putting it at the end of Luke 21, indicate that some scribe thought the pericope was of value.)
Per its most common location? John 8:2
At dawn he appeared again in the temple courts
Makes clear that, if not the author of the pericope, an early editor, was deliberately inserting it at this point.
Perhaps the original core of the pericope is just 8:3-11, with all of 7:53-8:2 being a transitional device. In that case, we have an independent pericope that could reflect any sort of "community." It could parallel John 4, with the "sinful" Samaritan woman, where Jesus brings up the issue of living water just like in John 7.
That said, how did it come to be attached to Luke in some cases?
The most interesting insertion point, though, is the mss that put it at the end of Luke 21. But, that's all triple tradition, and the idea that Luke would have made this addition doesn't make sense. But, although not specifically Lukan in style, the pericope seems to match him than John, or the other two Synoptics.
(This is also a refutation of fundagelical stances on inerrancy, of course, the idea that this was a "free-floating" pericope, valued by ante-Nicene communities but of uncertain provenance, so being stuck "somewhere." It's a refutation because it shows that scribes 1700-1800 years ago did NOT work and think that way.)
In addition, the one commenter citing the piece from Torah.com about Sukkoth at the time of the change of eras? That itself is perhaps somewhat dated in light of Yonathan Adler's new book, "The Origins of Judaism," though per Part 3 of my extended review, he does not talk too much about Sukkoth himself. On the other hand, the piece does well in picking up on Hyam Maccoby's contention that Jesus was entering Jerusalem at Sukkoth, not Passover. While the piece focuses on John, its insight is equally applicable to the Synoptics, both as far as the imagery and that of the three great festivals of Second Temple Judaism, it was Sukkoth that has the most Messianic tinges by far.
Anyway, even if John 8:2 is an original part of the pericope, the pericope as a whole, internally, has nothing to do with Sukkoth.
That said, it is nice to see "Religion Prof" self-outed as James McGrath.