  In this work we compute numerical bounds on the mass $\mu$ of superradiantly
unstable scalar fields in a Kerr black hole background using the continued
fraction method. We show that the normalized upper bound on the mass $\mu$
increases with the angular momentum number $\ell$ and the azimuthal number $m$,
approaching the most stringent analytical bound known to date when $\ell=m \gg
1$. We also provide an analytical fit to the numerically determined mass bound
as a function of the dimensionless spin parameter $a/M$ of the black hole with
an accuracy of the order $0.1\%$ for the fundamental mode with $\ell=m=1$, and
of the order $1\%$ for higher-order modes (up to $\ell=m=20$). We argue that
this analytical fit is particularly useful in astrophysical scenarios, since
the lowest $\ell=m$ modes are capable of producing the strongest observable
imprints of superradiance.
