We recently introduced a scaling theory for charge transport in molecular semiconductors with uncorrelated Gaussian energetic disorder, considering Miller-Abrahams as well as Marcus hopping and different lattice structures [ Cottaar et al. Phys. Rev. Lett. 107 136601 (2011)]. A compact expression was derived for the dependence of the charge-carrier mobility on temperature and carrier concentration. We show here that for Miller-Abrahams hopping the theory can also be applied to non-Gaussian energetic disorder, without parameter changes. Moreover, we show how it can be applied to correlated energetic disorder as obtained from randomly oriented molecular dipoles, which experiments suggest to be often more suitable. The same compact expression still describes the charge-carrier mobility, with new parameter values as determined from numerically exact results. The critical scaling exponent for correlated disorder is about twice as large as for uncorrelated disorder, which is caused by a different topology of the percolating network. The temperature dependence of the mobility for correlated disorder is significantly weaker than for uncorrelated disorder, while the carrier-concentration dependence is slightly weaker, due to small deviations of the density of states from a Gaussian. We indicate how comparison with experiments could distinguish between the different models.