In this paper we present the results of a generalized cluster\char22{}Bethe-lattice approach for the electronic density of states in binary alloys with cellular disorder, off-diagonal disorder, and short-range order. For application to large clusters, we suggest a new method for handling the problem of configuration averaging, in which we partition the cluster into a central "core" part and a surrounding layer. The "core" part is treated exactly, while the surrounding layer is described by a self-consistent complex self-energy $\ensuremath{\Sigma}(E)$, in determining which we take account of the fact that the atoms inside the cluster form a regular array. $\ensuremath{\Sigma}(E)$ therefore is not the same as the Bethe-lattice self-energy $\ensuremath{\sigma}(E)$, for which we take the coherent-potential-approximation value in the present application. In this multistage approach, we essentially adopt the philosophy of treating the different regions around a given site at different levels of approximation depending upon the importance of the region concerned for the local density of states at the central site. The feasibility of the present scheme is demonstrated through calculations with sets of model parameters. Of particular interest is the effect of off-diagonal disorder on the local density of states.