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Just searching around in Torat Emet finds a few mentions of different months, in Tanach, besides for Adar: Nissan Esther 3:7 Nechemiah 2:1 Sivan Esther 8:9 Tammuz Yechezkel 8:14 (kidding) Elul Nechemiah 6:15 Kislev Nechemiah 1:1 Zechariah 7:1 Teves Esther 2:16 Shevat Zechariah 1:7 NB: This list may be incomplete.
A different angle on the question: at the funeral for Rabbi Emanuel Gettinger (who was a mathematics and astronomy enthusiast), it was suggested that God values our engaging the world. This way, we would have to study astronomy to determine how to modify the calendar. (Recall that the Gemara praises the role of an honest human judge as "partnering with ...
As explained in this article the ideal Jewish year is actually 13 months (representing a dominance of the Jewish lunar calendar over the non-Jewish solar one) and it will resume this cycle in the days of Moshiach. The leap month is added to provide access to the future redemption even in our times.
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