C. F. Keil and Franz Delitzsch on Ezra 3:2

Ezr 3:2

Then the two leaders of the people, Joshua the high priest and Zerubbabel the prince (see on Jos 2:2), with their brethren, i.e., the priests and the men of Israel (the laity), arose and built the altar, to offer upon it burnt-offerings, as prescribed by the law of Moses, i.e., to restore the legal sacrifices. According to Ezr 3:6, the offering of burnt-offerings began on the first day of the seventh month; hence the altar was by this day already completed. This agrees with the statement, “When the seventh month approached” (Ezr 3:1), therefore before the first day of this month.