  We study the spin-$S$ Kitaev-Heisenberg model on the honeycomb lattice for
$S\!=\!1/2$, $1$ and $3/2$, by using the coupled cluster method (CCM) of
microscopic quantum many-body theory. This system is one of the earliest
extensions of the Kitaev model and is believed to contain two extended spin
liquid phases for any value of the spin quantum number $S$. We show that the
CCM delivers accurate estimates for the phase boundaries of these spin liquid
phases, as well as other transition points in the phase diagram. Moreover, we
find evidence of two unexpected narrow phases for $S\!=\!1/2$, one sandwiched
between the zigzag and ferromagnetic phases and the other between the N\'eel
and the stripy phases. The results establish the CCM as a versatile numerical
technique that can capture the strong quantum-mechanical fluctuations that are
inherently present in generalized Kitaev models with competing bond-dependent
anisotropies.
