Short answer: Yes, John was using the reference to the last day of the harvest feast to indicate that he was writing to people at the eschatological in-gathering.
The feast in view was originally a harvest festival. All Jewish men who were able to were required to come to Jerusalem and bring their first pickings which would in turn sanctify the rest of the harvest.
It celebrated the 40 years of wandering in the wilderness when the had escaped from Egypt and had not yet arrived at or entered the promised land yet. During this period the Jews would stay in temporary shelters in the desert and so this is celebrated by recreating the scene and staying in booths as well:
For an endearing portrait of the holiday I strongly recommend "Ushpizin", an Israeli film.
We know that John, seemingly at odds with the synoptics strongly links Jesus' death with the feast of the Passover:
[Jhn 19:14 KJV] (14) And it was the preparation of the passover, and about the sixth hour: and he saith unto the Jews, Behold your King!
I don't see where John mentions Pentecost but I believe the link in Acts of Pentecost with the beginning of the preaching of the acceptable year of the Lord is well established by Luke:
[Luk 4:18-19 KJV] (18) The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised, (19) To preach the acceptable year of the Lord.
[Act 2:7-21 KJV] (7) And they were all amazed and marvelled, saying one to another, Behold, are not all these which speak Galilaeans? (8) And how hear we every man in our own tongue, wherein we were born? (9) Parthians, and Medes, and Elamites, and the dwellers in Mesopotamia, and in Judaea, and Cappadocia, in Pontus, and Asia, (10) Phrygia, and Pamphylia, in Egypt, and in the parts of Libya about Cyrene, and strangers of Rome, Jews and proselytes, (11) Cretes and Arabians, we do hear them speak in our tongues the wonderful works of God. (12) And they were all amazed, and were in doubt, saying one to another, What meaneth this? (13) Others mocking said, These men are full of new wine. (14) But Peter, standing up with the eleven, lifted up his voice, and said unto them, Ye men of Judaea, and all ye that dwell at Jerusalem, be this known unto you, and hearken to my words: (15) For these are not drunken, as ye suppose, seeing it is but the third hour of the day. (16) But this is that which was spoken by the prophet Joel; (17) And it shall come to pass in the last days, saith God, I will pour out of my Spirit upon all flesh: and your sons and your daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams: (18) And on my servants and on my handmaidens I will pour out in those days of my Spirit; and they shall prophesy: (19) And I will shew wonders in heaven above, and signs in the earth beneath; blood, and fire, and vapour of smoke: (20) The sun shall be turned into darkness, and the moon into blood, before that great and notable day of the Lord come: (21) And it shall come to pass, that whosoever shall call on the name of the Lord shall be saved.
So there is every reason to think that John was "setting the stage" of the "harvest" just before the judgment.
Other NT authors link the first century to the period of Israel's being in the wilderness and the giving of the Torah:
[Mar 10:21 KJV] (21) Then Jesus beholding him loved him, and said unto him, One thing thou lackest: go thy way, sell whatsoever thou hast, and give to the poor, and thou shalt have treasure in heaven: and come, take up the cross, and follow me.
That was an allusion to:
[Deu 8:1-3 KJV] (1) All the commandments which I command thee this day shall ye observe to do, that ye may live, and multiply, and go in and possess the land which the LORD sware unto your fathers. (2) And thou shalt remember all the way which the LORD thy God led thee these forty years in the wilderness, to humble thee, and to prove thee, to know what was in thine heart, whether thou wouldest keep his commandments, or no. (3) And he humbled thee, and suffered thee to hunger, and fed thee with manna, which thou knewest not, neither did thy fathers know; that he might make thee know that man doth not live by bread only, but by every word that proceedeth out of the mouth of the LORD doth man live.
[1Co 10:1-11 NLT] (1) I don't want you to forget, dear brothers and sisters, about our ancestors in the wilderness long ago. All of them were guided by a cloud that moved ahead of them, and all of them walked through the sea on dry ground. (2) In the cloud and in the sea, all of them were baptized as followers of Moses. (3) All of them ate the same spiritual food, (4) and all of them drank the same spiritual water. For they drank from the spiritual rock that traveled with them, and that rock was Christ. (5) Yet God was not pleased with most of them, and their bodies were scattered in the wilderness. (6) These things happened as a warning to us, so that we would not crave evil things as they did, (7) or worship idols as some of them did. As the Scriptures say, "The people celebrated with feasting and drinking, and they indulged in pagan revelry." (8) And we must not engage in sexual immorality as some of them did, causing 23,000 of them to die in one day. (9) Nor should we put Christ to the test, as some of them did and then died from snakebites. (10) And don't grumble as some of them did, and then were destroyed by the angel of death. (11) These things happened to them as examples for us. They were written down to warn us who live at the end of the age.
[Heb 3:7-19 CSB] (7) Therefore, as the Holy Spirit says: Today, if you hear his voice, (8) do not harden your hearts as in the rebellion, on the day of testing in the wilderness, (9) where your fathers tested me, tried me, and saw my works (10) for forty years. Therefore I was provoked to anger with that generation and said, "They always go astray in their hearts, and they have not known my ways." (11) So I swore in my anger, "They will not enter my rest." (12) Watch out, brothers and sisters, so that there won't be in any of you an evil, unbelieving heart that turns away from the living God. (13) But encourage each other daily, while it is still called today, so that none of you is hardened by sin's deception. (14) For we have become participants in Christ if we hold firmly until the end the reality that we had at the start. (15) As it is said: Today, if you hear his voice, do not harden your hearts as in the rebellion. (16) For who heard and rebelled? Wasn't it all who came out of Egypt under Moses? (17) With whom was God angry for forty years? Wasn't it with those who sinned, whose bodies fell in the wilderness? (18) And to whom did he swear that they would not enter his rest, if not to those who disobeyed? (19) So we see that they were unable to enter because of unbelief.
[Heb 12:18-29 CSB] (18) For you have not come to what could be touched, to a blazing fire, to darkness, gloom, and storm, (19) to the blast of a trumpet, and the sound of words. Those who heard it begged that not another word be spoken to them, (20) for they could not bear what was commanded: If even an animal touches the mountain, it must be stoned. (21) The appearance was so terrifying that Moses said, I am trembling with fear. (22) Instead, you have come to Mount Zion, to the city of the living God (the heavenly Jerusalem), to myriads of angels, a festive gathering, (23) to the assembly of the firstborn whose names have been written in heaven, to a Judge, who is God of all, to the spirits of righteous people made perfect, (24) and to Jesus, the mediator of a new covenant, and to the sprinkled blood, which says better things than the blood of Abel. (25) See to it that you do not reject the one who speaks. For if they did not escape when they rejected him who warned them on earth, even less will we if we turn away from him who warns us from heaven. (26) His voice shook the earth at that time, but now he has promised, Yet once more I will shake not only the earth but also the heavens. (27) This expression, "Yet once more," indicates the removal of what can be shaken -- that is, created things -- so that what is not shaken might remain. (28) Therefore, since we are receiving a kingdom that cannot be shaken, let us be thankful. By it, we may serve God acceptably, with reverence and awe, (29) for our God is a consuming fire.**
In each case the promised land is just around the bend. And there are other examples.
This page has some information on how Christ fulfilled the various feasts and, because the feast of tabernacles is such an important one it gives many examples.
Originally the first day of the feast was the big day but later an eighth day was added and it became the great day:
...Furthermore, an eighth day is added as a concluding festival ("'aẓeret"; Lev. xxiii. 36, 39; Num. xxix. 35), which has an independent character: like the first day, it is a "holy convocation"; on it no labor is permitted, in which respect, as also in the sacrifices, it differs from the intervening six days. Exact sacrificial prescriptions are given in Num. xxix. 13-39, distinguishing this festival from the rest in the abundance of sacrifices. In addition to the daily regular offerings, there is to be a daily burnt offering of two rams, fourteen lambs, and bullocks—thirteen on the first day, twelve on the second day, and so diminishing by one daily on the other five days. In each case there is to be also the proper meal-offering of fine flour mixed with oil—three-tenths to each bullock, two-tenths to each ram, and one-tenth to each lamb. As a sin-offering a he-goat is to be sacrificed daily. On the eighth day the sacrifices differ, consisting of a he-goat as a sin-offering, and a single bullock, a ram, and seven lambs as a burnt offering—all of which are brought together with the meal-offering appropriate in each case...