  The superradiant instabilities of Kerr-Newman black holes with charged or
uncharged massive spin-0 fields are calculated analytically to the
next-to-leading order in the limit of $\alpha\sim r_g \mu \ll 1$. A missing
factor of $1/2$ in the previous leading-order result is identified. The
next-to-leading order result has a compact form and is in good agreement with
existing numerical calculations. The percentage error increases with $\alpha$,
from a few percent for $\alpha\sim 0.1$ to about $50\%$ for $\alpha\sim 0.4$.
Massive neutral scalars too heavy to be produced with Kerr black hole
superradiance may exist in the superradiant region of Kerr-Newman black holes.
