When magnetic skyrmions move under spin-orbit torque in magnetic nanowires, they experience a skyrmion Hall effect, which pushes them towards the nanowire edge where they risk being annihilated; this puts an upper limit on how fast they can be driven. However, the same magnetic multilayer harboring skyrmions can sustain a Neel-type strip domain wall along the nanowire length, potentially keeping the skyrmions separated from the edge. Here we study the interplay between current driven skyrmions and domain walls and find that they increase the annihilation current and allow the skyrmions to move faster. Based on the Thiele formalism, we confirm that the emergent longitudinal repulsive force and the modified energy landscape linked to the domain wall are responsible for the enhanced skyrmion motion. Furthermore, we identify that the longitudinal repulsive force emerges because of the broken axisymmetry in the local magnetization in front of the skyrmion. Our study uncovers key aspects in the interplay between two topological magnetic textures from different homotopy groups and may inspire new device concepts.