Simchas Torah A Once-a-Year Spiritual Opportunity

The Gemara says the days to call out to Hashem are the Aseres Yemei Teshuva, "when He is near." Yet the Zohar says Shemini Atzeres and Simchas Torah are an even better time to ask Hashem for anything at all. That seems backwards, since the Yamim Noraim just ended. What changed in these 21 days to make this moment closer still, and what should we be asking for? Rabbi Asaf Prisman closes with the last Simchas Torah of the Mir Yeshiva in its old city, when the students knew they might not get out, and one song they sang again and again.