Higher-order phases are characterized by corner or hinge modes that arise due to the interesting interplay of localization mechanisms along two or more dimensions. In this work, we introduce and construct a novel class of "hybrid" higher-order skin-topological boundary modes in nonreciprocal systems with two or more open boundaries. Their existence crucially relies on nonreciprocal pumping in addition to topological localization. Unlike usual non-Hermitian "skin" modes, they can exist in lattices with vanishing net reciprocity due to the selective nature of nonreciprocal pumping: While the bulk modes remain extended due to the cancellation of nonreciprocity within each unit cell, boundary modes experience a curious spontaneous breaking of reciprocity in the presence of topological localization, thereby experiencing the non-Hermitian skin effect. The number of possible hybridization channels increases rapidly with dimensionality, leading to a proliferation of distinct phases. In addition, skin modes or hybrid skin-topological modes can restore unitarity and are hence stable, allowing for experimental observations and manipulations in non-Hermitian photonic and electrical metamaterials.