In fact, Paul Verhoeven actually leans this way:1
Palm branches are typical of Sukkot, as they were brought to Jerusalem from the Jordan Valley in September and used to construct the tabernacles (skēnai). X. Léon-Dufour writes, "We can take it as assured that Jesus once visited Jerusalem in the autumn, at the feast of Tabernacles," and M. Goguel agrees. R. Brown concludes that "the theory that Jesus entered Jerusalem at Tabernacles rather than at Passover is interesting, but beyond the possibility of proof." I cannot prove it either, but I find it more likely that the triumphal entry took place during Sukkot than at the beginning of the Passion week. The euphoric mood of the scene is much more in keeping with Jesus' expectation that the kingdom of God was about to break through once and for all than the gloomy thought that he was going to die an excruciating death.
Some scholars (e.g., Smith, "No Time," 319) have argued that Jesus' triumphal entry actually occurred during Sukkot, partly because of the prominent role played in that holiday by the lûlāb, a bouquet of interwoven branches of palm, myrtle, and willow that is waved in festal processions (see m. Suk. 3:4 and cf. John 12:13, which specified that the Jerusalem crowds greeted Jesus with palm fronds—hence the designation "Palm Sunday"). Other Tabernacles associations of our passage and its surrounding context include the use of Psalm 118 ... and Jesus' expectation that he would find ripe figs on the tree in 11:13 (Tabernacles is a harvest festival).
David Freedman notes the association of palm branches with Tabernacles, but also refers to other occasions when they were used:3
Tabernacles came to be tied to Jewish hopes for a Davidic messiah and national independence. Judas Maccabees used the feast as a model for his celebration of the rededication of the Jerusalem altar in 164 B.C.E. (1 Macc. 4:54-59), the first celebration of Hannukah (2 Macc. 1:1-36). Elements of the Tabernacles celebration figure in Jesus' triumphal entry and crucifixion (Mark 11:1-11 par.), reflecting the people's longing for Jewish independence (cf. quotation from the Great Hallel, Ps. 118:25-26; the title Son of David, Matt. 21:9). The leaders of both Jewish revolts against Rome (66-70 and 132-135 C.E.) also used symbols and slogans from the feast.
On the basis of the final quotation provided above, it appears that Tabernacles 'imagery' and 'symbols' were being invoked out of season at least a couple of centuries before Jesus came along, for celebratory reasons, including in messianic contexts. While Judea was under Roman rule, palm branches were printed on their coins. During Jesus' day, it had become typical to use the psalm branch as a symbol for the Jewish nation.
It is certainly possible the passion narratives are compressing the timeline of Jesus' ministry for narrative purposes (it is already widely agreed that the Gospel authors do such a thing in other passages). However, such speculation isn't necessary. The concept that Jesus' entry into Jerusalem took place about a week before his death is unanimous in ancient Christian literature, including that of different traditions (as John is apart from Mark, Matthew, and Luke). Hence, it is equally possible (and more historically probable, given the independent attestation), that in the wake of a messianic movement centered around Jesus, residents of Jerusalem welcomed him into the capital city using a common symbol for their nation.
1 Paul Verhoeven, Jesus of Nazareth. (Yes, that Paul Verhoeven. The film director is a member of the Jesus Seminar.)
2 Joel Marcus, Mark 8-16: A New Translation with Introduction and Commentary.
3 David Noel Freedman, Allen C. Myers, Eerdmans Dictionary of the Bible.