(לח) וַיֵּ֣רֶד צָד֣וֹק הַ֠כֹּהֵן וְנָתָ֨ן הַנָּבִ֜יא וּבְנָיָ֣הוּ בֶן־יְהוֹיָדָ֗ע וְהַכְּרֵתִי֙ וְהַפְּלֵתִ֔י וַיַּרְכִּ֙בוּ֙ אֶת־שְׁלֹמֹ֔ה עַל־פִּרְדַּ֖ת הַמֶּ֣לֶךְ דָּוִ֑ד וַיֹּלִ֥כוּ אֹת֖וֹ עַל־גִּחֽוֹן׃
(38) So Zadok the priest, and Nathan the prophet, and Benaiah the son of Jehoiada, and the Cherethites and the Pelethites, went down, and caused Solomon to ride upon king David’s mule, and brought him to Gihon.
(לט) וַיִּקַּח֩ צָד֨וֹק הַכֹּהֵ֜ן אֶת־קֶ֤רֶן הַשֶּׁ֙מֶן֙ מִן־הָאֹ֔הֶל וַיִּמְשַׁ֖ח אֶת־שְׁלֹמֹ֑ה וַֽיִּתְקְעוּ֙ בַּשּׁוֹפָ֔ר וַיֹּֽאמְרוּ֙ כָּל־הָעָ֔ם יְחִ֖י הַמֶּ֥לֶךְ שְׁלֹמֹֽה׃ (מ) וַיַּעֲל֤וּ כָל־הָעָם֙ אַֽחֲרָ֔יו וְהָעָם֙ מְחַלְּלִ֣ים בַּחֲלִלִ֔ים וּשְׂמֵחִ֖ים שִׂמְחָ֣ה גְדוֹלָ֑ה וַתִּבָּקַ֥ע הָאָ֖רֶץ בְּקוֹלָֽם׃
(39) And Zadok the priest took the horn of oil out of the Tent, and anointed Solomon. And they blew the ram’s horn; and all the people said: ‘Long live king Solomon.’ (40) And all the people came up after him, and the people piped with pipes, and rejoiced with great joy, so that the earth rent with the sound of them.
This shows how King Soloman got anointed as king and this shows how they anointed kings in the times of the Navi.
(ג) וַ֠יָּבֹאוּ כָּל־זִקְנֵ֨י יִשְׂרָאֵ֤ל אֶל־הַמֶּ֙לֶךְ֙ חֶבְר֔וֹנָה וַיִּכְרֹ֣ת לָהֶם֩ הַמֶּ֨לֶךְ דָּוִ֥ד בְּרִ֛ית בְּחֶבְר֖וֹן לִפְנֵ֣י יי וַיִּמְשְׁח֧וּ אֶת־דָּוִ֛ד לְמֶ֖לֶךְ עַל־יִשְׂרָאֵֽל׃ (פ)
(3) So all the elders of Israel came to the king to Hebron; and king David made a covenant with them in Hebron before the LORD; and they anointed David king over Israel.
You can see that David and Soloman got anointed in different ways.
