In John 5, there is a story which takes place after "a feast of the Jews." What feast was this?
After this there was a feast of the Jews, and Jesus went up to Jerusalem.
There are three holidays that include the obligation to go up to Jerusalem, Passover, Weeks and Tabernacles. Other holiday such as the New Year (Rosh HaShanah), Atonement (Yom Kipur) and Dedication (Hanukah) do not include this obligation. The obligation is given in Deut 16:16-17:
שלוש פעמים בשנה יראה כל זכורך את פני יהוה אלהיך במקום אשר יבחר בחג המצות ובחג השבעות ובחג הסכות ולא יראה את פני יהוה ריקם. איש כמתנת ידו כברכת יהוה אלהיך אשר נתן לך
Passover is in the early spring. Weeks is in the early summer and Tabernacles is in the early fall.
Assuming that the narratives in John 4-6 are in chronological order, and assuming that John 4:35, "four months more to the harvest" was said at a time of four months before the wheat harvest (early summer) then the festival mentioned in John 5:1 could be either Passover, Weeks or Tabernacles.
It could also be that "four months more to the harvest" refers not to the wheat harvest but to the general harvest time in the early fall and therefore is a reference to Tabernacles, which is called "the harvest holiday" (Exodus 23:16), and to the preceding forty days of repentance, the month of Elul and beginning of Tishrei that end with the Day of Atonement, Yom Kipur, just before Tabernacles. Then the message in John 4:35 is clear. He is saying, don't wait for the month of Elul to start to repent, repent now.
The fact that some witnesses read "for the Jewish festival", in John 5:1 suggests that the festival was Tabernacles, because Tabernacles is the only Jewish festival that is commonly called simply "חג" or "the festival". (Kings I 8:2, Nehemiah 8:14, and Babylonian Talmud tractate Rosh Hashanah 6a). Other holidays are always referred to by their particular names.
The next reference to seasons that we have is John 6:4, "...near Passover time". Assuming that events narrated are sequential and in the same year, then this would suggest that the previous festival with an obligation to go up to Jerusalem mentioned in John 5:1 was Tabernacles.
So, there are good reasons to surmise that John 5:1 refers to Tabernacles, though we need to make some brave assumptions about the text to do this.