Note that in I Shmuel 10:1
וַיִּקַּח שְׁמוּאֵל אֶת פַּךְ הַשֶּׁמֶן וַיִּצֹק עַל רֹאשׁוֹ
וַיִּשָּׁקֵהוּ וַיֹּאמֶר הֲלוֹא כִּי מְשָׁחֲךָ יְהֹוָה עַל נַחֲלָתוֹ
And Samuel took the vial of oil, and poured it on his head, and kissed
him. And he said, "Indeed, the Lord has anointed you to be a ruler
over His inheritance.
This implies that it was the special anointing oil.
Rambam in Hilchos Melachim 1:7 says
When a king is appointed, he is anointed with oil reserved for this
purpose, as I Samuel 10:1 states: 'And Samuel took the cruse of oil
and poured it over his head. Then, he kissed him.'
However,once David Hamelech was anointed, the special oil was only used for the Davidic kings and only if there was a dispute as to who would be the king.
Kings of Israel are not anointed with the special anointing oil, but
with Afarsimon oil. Only a descendent of David may be appointed as
king in Jerusalem. And only descendents of David are anointed with the
special anointing oil.
A son who succeeds his father as king is not anointed unless he
assumes his position amid a dispute over the inheritance or during a
civil war. Under these circumstances, he should be anointed in order
to remove all disagreement.
Therefore, they anointed Solomon because of the claim of Adoniyahu,
Jehoash, because of the usurpation of Atalyah, and Jehoachaz, because
of [the claim of his brother, Jehoyakim.