Spin Hall currents generated in nonmagnetic materials are conventionally regarded as bulk responses whose polarization is fixed by crystal symmetry. This view has motivated the search for intrinsically low-symmetry spin sources when unconventional spin polarizations are required. Here we point out that, in realistic heterostructures, the device-relevant quantity is not the fully symmetry-averaged bulk spin Hall current, but the emitted spin current transmitted across the interface. We therefore establish emitted spin currents as bulk-interface hybrid responses and propose interfacial noncollinear filtering as a mechanism to bypass the bulk-symmetry constraint. A low-symmetry interfacial spin-orbit field, generally noncollinear with the momentum-resolved spin polarization of the incident spin Hall current, imposes spin-dependent transmission and converts hidden momentum-resolved spin-polarization components into an observable unconventional emitted spin current. Using both a rotationally symmetric minimal model and a realistic high-symmetry Dirac-semimetal model, we show that conventional spin Hall sources can emit sizable out-of-plane spin currents when their hidden bulk spin Hall textures are selectively transmitted by the interfacial spin-orbit field. Our results reveal that spin-current polarization emerges from the cooperative action of bulk and interfacial responses, providing a strategy for reprogramming spin-current polarization in high-efficiency, CMOS-compatible spin Hall materials without relying on intrinsically low-symmetry bulk crystals or external symmetry-breaking schemes.