Pey - Payis, a new raffle for the Korbonos of Shmini Atzeres that don't go to the Mishmoros who didn't have 3 Parim on Succos.
Zayin - Zman, a new Shehecheyanu
Reish - Regel. It is a machlokes what this means. Either that you don't sit in a Succah (Rashi), or that yo0u must stay over night, "Lina" (Rabbeinu Tam), or it stops aveilus (Rabbeinu Chananel).
Korbon - Korban A different Korban and not a continuation of the steady decline of Parim from Succos.
Shin - Shira Shir Shel Yom which is not connected to the theme of the Succos days.
Bais - Bracha. It is a machlokes what this means. Either that you bless the king (Rashi), or that you say "Shmini Atzeres" in davening and Birchas HaMazon (Rabbeinu Tam)
See Succah 48a