  We present new equilibrium solutions of stationary models of magnetized thick
disks (or tori) around Kerr black holes with synchronised scalar hair. The
models reported here largely extend our previous results based on constant
radial distributions of the specific angular momentum along the equatorial
plane. We introduce a new way to prescribe the distribution of the disk's
angular momentum based on a combination of two previous proposals and compute
the angular momentum distribution outside the equatorial plane by resorting to
the construction of von Zeipel cylinders. We find that the effect of the scalar
hair on the black hole spacetime can yield significant differences in the disk
morphology and properties compared to what is found if the spacetime is purely
Kerr. Some of the tori built within the most extreme, background hairy black
hole spacetime of our sample exhibit the appearance of two maxima in the
gravitational energy density which impacts the radial profile distributions of
the disk's thermodynamical quantities. The models reported in this paper can be
used as initial data for numerical evolutions with GRMHD codes to study their
stability properties. Moreover, they can be employed as illuminating sources to
build shadows of Kerr black holes with scalar hair which might help further
constrain the no-hair hypothesis as new observational data is collected.
