  In this paper we first investigate the equatorial circular orbit structure of
Kerr black holes with scalar hair (KBHsSH) and highlight their most prominent
features which are quite distinct from the exterior region of ordinary bald
Kerr black holes, i.e. peculiarities that arise from the combined bound system
of a hole with an off-center, self-gravitating distribution of scalar matter.
Some of these traits are incompatible with the thin disk approach, thus we
identify and map out various regions in the parameter space respectively. All
the solutions for which the stable circular orbital velocity (and angular
momentum) curve is continuous are used for building thin and optically thick
disks around them, from which we extract the radiant energy fluxes,
luminosities and efficiencies. We compare the results in batches with the same
spin parameter $j$ but different normalized charges, and the profiles are
richly diverse. Because of the existence of a conserved scalar charge, $Q$,
these solutions are non-unique in the $(M, J)$ parameter space. Furthermore,
$Q$ cannot be extracted asymptotically from the metric functions. Nevertheless,
by constraining the parameters through different observations, the luminosity
profile could in turn be used to constrain the Noether charge and characterize
the spacetime, should KBHsSH exist.
