Why is the Rosh Hashana for kings not the same Rosh Hashana of the year when we coronate the King of Kings? See Rosh Hashana (1:1)
בְּאֶחָד בְּנִיסָן רֹאשׁ הַשָּׁנָה לַמְּלָכִים וְלָרְגָלִים. בְּאֶחָד בֶּאֱלוּל רֹאשׁ הַשָּׁנָה לְמַעְשַׂר בְּהֵמָה. רַבִּי אֶלְעָזָר וְרַבִּי שִׁמְעוֹן אוֹמְרִים, בְּאֶחָד בְּתִשְׁרֵי. בְּאֶחָד בְּתִשְׁרֵי רֹאשׁ הַשָּׁנָה לַשָּׁנִים וְלַשְּׁמִטִּין וְלַיּוֹבְלוֹת, לַנְּטִיעָה וְלַיְרָקוֹת.
The four new years are: On the first of Nisan, the new year for the kings and for the festivals; On the first of Elul, the new year for the tithing of animals; Rabbi Eliezer and Rabbi Shimon say, on the first of Tishrei. On the first of Tishrei, the new year for years, for the Sabbatical years and for the Jubilee years and for the planting and for the vegetables.