The Chumash refers to the months simply as "the first", "the second", and so on. Today we know them by their Babylonian names (Nisan, Iyar ...) Occasionally the Prophets from the First-Temple Era use older Hebrew (?) names, such as "Ziv."
Do we have a complete list of those names for the months?
In Tanach there are only four:
In a letter from the era of the first Beis Hamikdash found in Arad, there is mention (according to some reconstructions) of ירח צח, "the month Tzach." So it has been proposed that Isaiah 18:4, כחם צח...עלי קציר, is referring to that month (which would mean that it corresponds to Sivan, harvest time).
Other than that, there's the Gezer Calendar, although that just gives agricultural activities for each month rather than names.
Scholars believe that we shared calendar names with the Phoenicians (Tzidonim in Tanach) (see here, pg. 76, for example), or that perhaps we even borrowed the names from them (seeing as there aren't month names in the Torah, only from the time of Shlomo, who had close ties with the Phoenicians (Dr. Chagai Misgav, for example, holds the latter view)).
If this is true, then we can look at the rest of the Phoenician months to know what names more or less were used in Israel (list taken from Prof. Robert Stieglitz's essay "The Phoenician-Punic Menology" (can be read in preview mode here)) (note that the names were written in Phoenician, which is a Hebrew dialect with some differences between Israelite Hebrew and itself):
זבח שמש - Zevach Shemesh - Probably Sivan
חיר - Khir - Probably Adar (note the name resembles that of the king of Tzor (Tzidon), Khiram)
פגרם - Pegarim - Probably Tevet (may have been a reference to cultic stelae (see here))
מפע - Mofiah - Probably Elul
מפע לפני - Mofiah Lifnei - either an intercalary month (חודש מעובר) like Adar Sheni, or perhaps parallel to Av (and then it's literally "lifnei" (before) Mofiah)
מרפא/מרפאם - Merapeh/Merapim - Probably Kislev
מתן - Matan - Probably Nissan
פעלת - Pe'ulot/Po'elet - Probably Shevat
כרר - Karar - Probably Tammuz
Personally, I'm skeptical that Am Yisrael ever used these names themselves, seeing as the Tanach has to tell the reader to which Hebrew month each name corresponds to:
"...in the month of Ziv—that is, the second month..."
"...in the month of Bul—that is, the eighth month..."
"...in the month of Ethanim—that is, the seventh month..."
Moreover, the other names have explicit or implicit idolatrous connotations: Zevach Shemesh - sacrifice to the sun; Khir - per the name Khiram, a Phoenician king, which appears to mean "Khir is great" (rather than חי-רם), it seems that Khir was a Phoenician diety; Pegarim - probably means cultic stele (see above); Mofiah - probably refers to the appearance of some sort of deity; Mofiah Lifnei - probably same; Merapim - multiple deities healing mankind or from Baal Merapeh, a Canaanite deity according to C.R. Conrad, PEQS 21.1; Matan - either referring to gifts given to the Phoenician gods, or referring to Baal priests (comp. Melachim 2:11:18 and Divrei Hayamim 2:23:17; Izevel princess of Tzidon brought over Baal worship to Israel and Ataliyahu brought it over to Judea); Pe'ulot/Po'elet - either multiple acts of deities, or a female deity's acts - Ashtoret, perhaps; Karar - possibly from "Kar" (כר) which means "sheep" in Hebrew, and therefore refers either to idolatrous sacrifices, or to deities with livestock motifs, or perhaps the name of a deity (there were people named Evedkarar).
Therefore, I think only a select few months were loaned over temporarily, just for the duration of the construction of the Temple, specifically, the names with neutral connotations: Ziv - splendor; Bul - clump; Ha'Eitanim - apparently from a Hurrian word that means "fathers", which ties in with the gemara in the Bavli Rosh Hashanah 11a and the Yerushalmi Rosh Hashanah 1:2 that say that this is the month of the Avot. Why would that be the case, that these three names were loaned over temporarily, I'm not really sure yet.
The book of Gad the Seer contains few more names, unfortunately , I can't remember which.
Also, this book is authenticity is questionable, so take it with caution.
