  We continue to explore the bootstrap approach to five-point correlation
functions for IIB supergravity on $AdS_5\times S^5$. Building on the result of
[1], we develop an improved algorithm that allows us to more efficiently
compute correlators of higher Kaluza-Klein modes. The new method uses only
factorization and a superconformal twist, and is entirely within Mellin space
where the analytic structure of holographic correlators is simpler. Using this
method, we obtain in a closed form all five-point functions of the form
$\langle pp222\rangle$, extending the earlier result for $p=2$. As a byproduct
of our analysis, we also obtain explicit results for spinning four-point
functions of higher Kaluza-Klein modes.
