Reduced-order particle-in-cell (PIC) scheme is a novel kinetic modeling method for plasma systems that has been developed at Imperial Plasma Propulsion Laboratory. Since its conception, the scheme has become progressively more mature and has undergone a rigorous step-by-step verification to ensure its generalizability, accuracy, and reliability. In this article, we expand our verification efforts to further ascertain the applicability and versatility of the reduced-order PIC and its underpinning dimensionality-reduction approach in new simulation configurations and test cases. We first present the verification results against a recent Penning discharge benchmark case. This test case assesses the ability of the reduced-order PIC to resolve the underlying physics in a setup where the principal direction of the involved phenomena is constantly changing with respect to the coordinate system. Second, we present results from reduced-order quasi-2D (Q2D) axial-radial Hall thruster simulations with various approximation orders of the 2D problem. These results complement the verifications carried out so far in the axial-azimuthal and radial-azimuthal Hall thruster coordinates. As a precursor to the Q2D code verification in the axial-radial coordinates, we demonstrate that the methodology underlying the reduced-order PIC can also enable reduced-order Direct Simulation Monte Carlo studies for the neutral gas dynamics. This is crucial because: (1) a kinetic-DSMC neutral dynamics module is an essential component of Hall thruster PIC simulations in axial-radial geometries and, (2) this module needs to be compatible with the reduced-order definition of the PIC simulation itself. The latest verifications and tests reported in this work demonstrated that the reduced-order PIC and its enabling underlying dimensionality-reduction technique are fully versatile and generalizable. This paves the way toward reliably extending the method to 3D to achieve a quasi-3D PIC code that realizes cost-efficient modeling of real-size E×B plasma technologies.