The Shabbos after yomtov is often referred to as “Shabbos nach yomtov.” The Divrei Shmuel of Slonim zy”a said in the name of Rav Dovid Moshe of Tchortkov zy”a that this expression means that it’s still yomtov (nach yomtov), meaning that it still has the sanctity of yomtov.
The Divrei Shmuel adds that the Tchotkover said this about the Shabbos after Shavuos, but it applies even more so to the Shabbos after Simchas Torah, as everyone agrees that the Torah was given on Shabbos (Shabbos 86B), and all the good influences that flow to this world are in the merit of the Torah and Shabbos.