We investigated how induced optical bistability (IOB) and bistability domain (BD) of spheroidal core–shell nanocomposites (NCs) are affected by the depolarization factor (L), metal fraction (p), and host matrix (ɛh). The NCs are composed of passive and active dielectric cores coated with a metallic shell and embedded in various host matrices. By employing Laplace’s equation and the Drude–Lorentz model, we calculated the electric field enhancement in various regions of the NCs and studied their IOB theoretically and numerically. The BD of the same NCs is investigated using the root analysis approach. The study reveals that when the value of L decreases, the bistable region of OB increases. However, when p increases at constant L and ɛh, the bistable region gets wider. When the value of ɛh increases, the IOB region is achieved at larger values of incident field. Moreover, the IOB produced was narrower in the passive dielectric core than in the active one, showing that the type of core material also influences the IOB of the core–shell NCs. When the value of L changes, the region of BD also changes in both types of cores. Similarly, changing the value of p in both types of cores causes the BD to vanish or emerge. Moreover, varying ɛh, BD is produced in the OB regions. Overall, the IOB and BD in each core are sensitive to changes in L, p, and ɛh. Varying these parameters leads to the possibility of tuning the bistable regions, which can be used in optical circuits, logic operations, and optical memory.