In plasma edge simulations using the SOLPS-ITER code, the simulated Scrape-Off Layer plasma domain has historically been restricted to magnetic flux surfaces contacting divertor targets at both ends. We present here a newly developed numerical solver for the B2.5 plasma solver in SOLPS-ITER, allowing the numerical grid to be extended to the true vessel boundaries. The new, unstructured Finite Volume scheme can deal with arbitrary grids and magnetic topologies in the 2D poloidal plane. It includes a correct numerical treatment of possibly misaligned faces and cells w.r.t. the magnetic field to cope with, for example, strong divertor target shaping. The solver combines the benefits of an accurate numerical separation of fast parallel and slow radial transport, with a realistic description of the wall geometry, and the possibility of local grid refinement to capture sharp features in the Scrape-Off Layer flows. Generalized sheath boundary conditions are presented that can be imposed at all vessel boundaries, removing an important modeling uncertainty related to the specification of ad hoc decay length boundary conditions at the outer flux surfaces. The resulting model is applied to an AUG single-null case, a standard benchmark case for SOLPS-ITER. We analyze in particular the impact of the extended plasma model on upstream and divertor plasma conditions, and the improved predictions of heat and particle loads to the main chamber wall. The extended solver also allows for a much improved qualitative agreement between fluid and kinetic neutral simulations, because the fluid neutral solution, which is obtained on the plasma grid, now also extends to the true main chamber and divertor vessel boundaries.