Teshuvah draws down the sukkah's radiance

מצות הסוכה אחר ימי תשובה.

The mitzvah of Sukkah comes after the days of teshuvah.

The Sefas Emes notes the calendar: Sukkos follows directly upon Rosh Hashanah and Yom Kippur, and this order is itself meaningful.

עפ"י הגמ' דסוכה בפסוק מאחז פני כסא פרשז עליו עננו פירש מזיו שכינתו עליו.

This is based on the Gemara in Sukkah, on the verse "He grasps the face of the throne, He spreads His cloud over it" (Iyov 26:9) — He spread from the radiance of His Shechinah upon it.

Chazal read the word "parshez" as a contraction (pireish, ziv): Hashem spreads the glow of His Presence over the throne, and this spreading-out of the Shechinah is the inner root of the sukkah.

וזה הענין מתקיים בכללות בנ"י.

And this matter is fulfilled in the collective body of Bnei Yisrael.

What happens by the heavenly throne is mirrored down in the nation as a whole.

שעושין תשובה ואמרו חז"ל שמגעת עד כסא הכבוד.

For they do teshuvah, and Chazal said that teshuvah reaches all the way to the Throne of Glory.

The teshuvah of the Days of Awe ascends to the very kisei hakavod (Throne of Glory) — exactly the place over which the radiance is spread.

לכן יורד אח"כ הארה מזיו כסא כבודו וסוכך.

Therefore afterward a radiance descends from the glow of the Throne of His Glory and forms a covering.

Because the teshuvah rose to the throne, in return a light comes down from that very throne and "spreads over" us as a sukkah.

וגונן עלינו.

And it shields us.

That descending radiance becomes a protective canopy over Bnei Yisrael.

כאשר חכמים הגידו שחל שם שמים על הסוכה.

As the Chachamim told us, that the Name of Heaven rests upon the sukkah.

The sukkah is not merely a hut; the very Name of Hashem rests upon it, which is why it carries such sanctity.

לכן צריך להיות למעלה מי' ע"ש בגמ' הנ"ל ואמרם אשתרביב לי' כסא כו'.

That is why it must be above ten tefachim — see there in the aforementioned Gemara, and their statement that "the throne was extended down to him."

The halachah that the sukkah's covering be above ten handbreadths is tied to this idea: the throne (whose lowest reach is ten) was stretched downward, so the sukkah must reach into that domain where the radiance of the throne dwells.

וזה הענין הי' ג"כ בזמן המקדש שנכנס הכה"ג לפני ולפנים ביוה"כ בענן הקטורת.

And this same matter occurred too in the time of the Mikdash, when the Kohen Gadol entered the innermost chamber on Yom Kippur amid the cloud of the ketores.

The pattern repeats: on Yom Kippur the Kohen Gadol penetrated to the Holy of Holies enveloped in the incense cloud — itself a kind of sukkah of cloud at the height of teshuvah.

לכן המשיך הארה שישרה ענני כבודו על בנ"י אח"כ בימי החג.

Therefore he drew down a radiance so that His clouds of glory would rest upon Bnei Yisrael afterward, in the days of the festival.

That avodah of Yom Kippur drew down the light that would later settle as the Clouds of Glory upon the people during Sukkos.

ועיין בזוה"ק אמור ק"ג א' שמביא הפסוק וכסה ענן הקטורת למצות סוכה:

And see the Zohar HaKadosh, Emor 103a, which connects the verse "and the cloud of the ketores shall cover" (Vayikra 16:13) to the mitzvah of sukkah.

The Zohar makes the link explicit: the covering cloud of the incense in the Holy of Holies and the s'chach of the sukkah are one idea — a canopy of holy concealment within which the Shechinah rests.

Summary: Sukkos comes right after the Days of Awe because the teshuvah of Bnei Yisrael rises to the Throne of Glory, and in return a radiance descends from that throne and spreads over the nation as the sukkah — the very Name of Hashem resting upon it. This mirrors the Kohen Gadol entering the Holy of Holies in the cloud of ketores on Yom Kippur, which drew down the Clouds of Glory that become the sukkah of the festival.