  Considering the large-$q$ expansion of the Sachdev-Ye-Kitaev (SYK) model in
the two-stage limit, we compute the Lanczos coefficients, Krylov complexity,
and the higher Krylov cumulants in subleading order, along with the $t/q$
effects. The Krylov complexity naturally describes the "size" of the
distribution, while the higher cumulants encode richer information. We further
consider the double-scaled limit of SYK$_q$ at infinite temperature, where $q
\sim \sqrt{N}$. In such a limit, we find that the scrambling time shrinks to
zero, and the Lanczos coefficients diverge. The growth of Krylov complexity
appears to be "hyperfast", which is previously conjectured to be associated
with scrambling in de Sitter space.
