The interplay between the spin-orbit and Zeeman interactions acting on a spinful Su-Schrieffer-Heeger model is studied based on an InAs nanowire subjected to a periodic gate potential along the axial direction. It is shown that a nontrivial topological phase can be achieved by regulating the confining-potential configuration. In the absence of the Zeeman field, we prove that the topology of the chain is not affected by the Rashba spin-orbit interaction due to the persisting chiral symmetry. The energies of the edge modes can be manipulated by varying the magnitude and direction of the external magnetic field. Remarkably, the joint effect of the two spin-related interactions leads to novel edge states that appear in the gap formed by the anticrossing of the bands of a finite spinful dimerized chain, and can be merged into the bulk states by tilting the magnetic-field direction.