  We elaborate on a new recursive method proposed in arXiv:2312.15484 for
computing tree-level $n$-point supergluon amplitudes as well as those with one
gluon, i.e., spinning amplitudes, in ${\rm AdS}_5 \times S^3$. We present an
improved proof for the so-called "constructibility" of supergluon and spinning
amplitudes based on their factorizations and flat-space limit, which allows us
to determine these amplitudes in Mellin space to all $n$. We present explicit
and remarkably simple expressions for up to $n=7$ supergluon amplitudes and
$n=6$ spinning amplitudes, which can be viewed as AdS generalizations of the
scalar-scaffolded gluon amplitudes proposed recently. We then reveal a series
of hidden structures of these AdS amplitudes including (1) an understanding of
general pole structures especially the precise truncation on descendent poles
(2) a derivation of simple "Feynman rules" for the all-$n$ amplitudes with the
simplest R-symmetry structures, and (3) certain universal behavior analogous to
the soft/collinear limit of flat-space amplitudes.
