  Kerr black holes with synchronised scalar hair and azimuthal harmonic index
$m>1$ are constructed and studied. The corresponding domain of existence has a
broader frequency range than the fundamental $m=1$ family; moreover, larger ADM
masses, $M$ and angular momenta $J$ are allowed. Amongst other salient
features, non-uniqueness of solutions for fixed global quantities is observed:
solutions with the same $M$ and $J$ co-exist, for consecutive values of $m$,
and the ones with larger $m$ are always entropically favoured. Our analysis
demonstrates, moreover, the qualitative universality of various features
observed for $m=1$ solutions, such as the shape of the domain of existence, the
typology of ergo-regions, and the horizon geometry, which is studied through
its isometric embedding in Euclidean 3-space.
