Moshiach must definitionally descend from David, and according to this answer he must also descend from Shlomo. However, Moshiach cannot descend from Yehoaichin because of Yirmiyahu's curse.
Are there any other kings of Judah from which Moshiach must descend?
The curse of Yechonya (or Yehoyachin) was revoked and his wife had two sons while he was in prison. This is rebuilding of the Davidic line through Zerubavel. Note that this means that since Yehoyachin was the last king who had children that survived (all of Tzidkiyahus children were killed at the destruction of the first temple) Mashiach would therefore continue in a direct line from King David through every king of the House of David through Yechonya to Zerubavel. If anyone is a descendant outside of that direct line, then he would not be eligible to become the Mashiach.
R. Zeira said,
I heard something with regard to this which R. Shmuel b. R. Yitzchak expounded, but I forgot what it was.
R. Acha Aricha said to him,
Perhaps this is (the verse he expounded)
“Thus says the Lord: Write you this man childless,
A man who shall not succeed in his days.”
(R. Zeira) said, yes (that is the verse he said) “in his days” he shall not succeed, in the days of his son he shall succeed.
R. Acha and R. Avin b. Binyamin in the name of R. Abba:
Great is the power of repentance, for it cancels out both the (evil) decree and the oath.
The oath, as it says, “As I live - declares the Lord - if you O Conyahu, son of Yehoyakim of Judah were a signet ring on my right hand, I would tear you off even there” (Jer. 22:24).
And cancels the decree:
“Thus says the Lord: Write you this man childless.”
Yet latter it is written: “And the sons of Yechonya, Asir his son, She’altiel his son” (I Chron. 3:17)
‘Asir his son’
because he was in prison (beit ha-asurim);
‘She’altiel his son’
Because from him the Davidic line was replanted.
R. Tanchum b. R. Yirmiyah said,
‘Asir’ –
This is the Holy One Blessed Be He,
Who bound (asar) himself with an oath.]
‘She’altiel’ -
That He asked (sh’aal) the court in Heaven
(to release Him from) His oath.
This shiur continues that the mashiach must be a direct descendant of all the kings (because it must go through Yechonya).
Before proceeding to the next section of the story, it is worth noting that the notion that the royal line of David flows exclusively through Yechonya is already suggested by the verses in I Chronicles 3. Verses 10 through 14 list the kings of Judah from Solomon to Yoshiyahu. Verse 15 lists the four sons of Yoshiyahu, including Yehoyakim. Verse 16 lists the two sons of Yehoyakim, Tzidkiya and Yechonya. Following this, the verses to the end of the chapter go on to record several generations of Yechonya’s descendants, presumably down to the time of the writing of Chronicles. This genealogy suggests that the royal line of David is continued only by Yechonya. The implication is that had he not had children, David’s line would have come to an end.
I also found The Three Pronged Message of Chagai, Zechariah, and Malachi and Its Relevance for Our Times - Part 3 by Rabbi Chaim Jachter which also seems to point to Zerubavel as the melech, as well as Ezra-Nechemia - The Most Relevant Sefer in Tanach - Part 1 which also points to Zerubavel as the next melech.
Rav Yoel Bin Nun argues that there is another crisis that Sefer Ezra quietly and implicitly teaches us how to manage. This crisis revolves around the failure of Zerubavel to emerge as the Mashiach. The first six Perakim of Sefer Ezra speak of the years that Zerubavel served as governor of Eretz Yisrael on behalf of the Persian Empire. It seems that the Jews had great expectations of Zerubavel. First, it is important to note that Zerubavel was the great grandson of Yehoyachin (Yechanya), the last king of Judea during the period of Bayit Rishon (Divrei Hayamim 3:16-19). Thus, the Persians empowered a Jew who was part of the Davidic line (and thus potentially the Mashiach) to govern Eretz Yisrael. This presumably raised the hopes among Jews that the glory days of Bayit Rishon could be restored and even improved upon.
Second, both Chaggai (2:21-23) and Zechariah (6:12-15) seem to prophesize that Zerubavel had the potential to become Mashiach, or at least to restore Jewish sovereignty over Eretz Yisrael. Unfortunately, these promises did not materialize (their fulfillment was delayed until the time of Hasmoneans and beyond) and must have left the Jews in Eretz Yisrael profoundly demoralized. Indeed, a sense of crisis is expressed in Nechemia 9:37 (see the broader context), where it is stated, “UVeTzarah Gedolah Anachnu,” “We are in great distress.” Also see Ezra 9:8-9 where similar emotions are expressed.
Abarbanel on Zechariah 6:12 says that the crown of the kingdom belongs to Zerubavel and his descendants, but he himself will not be crowned as he is a "shoot", not yet grown. Metzudas David says that the Mashiach will be descended from Zerubavel. Similarly on Chagai 2:23
There is a web site that discusses the lines of descent from Zerubavel through the ages.
Descendants of Zerubavel Rashei Galut (Exilarchs), Gaonim and Rabbis. The page is too big to copy here, so just follow the link.
According to the Zohar (Shelach 173b), the Messiah will descend from Hefzibah, the wife of Nathan (son of King David). Philo understands that Joash was a descendant of Nathan, son of David and the Messiah will descend from him and not from King Solomon. But Chronicles I 3:11 says that Joash was from the line directly descending from Kin Solomon. In some places, Maimonides writes that the Messiah will be a descendant of King David and King Solomon, but in other places he writes just King David, not King Solomon. Midrash Tanchuma (Toldos) says that the Messiah will be a descendant of Zerubabel.
