The potential for post-acceleration of relativistic electrons in vacuum with the help of monocyclic THz electromagnetic pulses augmented with constant magnetic fields is explored. The directions of the electromagnetic wave propagation and electron injection are completely or closely aligned to enhance the time of energy transfer from the field to the electron. Monocyclic THz pulses efficiently capture and drive fast electrons until they are released due to field-induced transverse displacement or overtaken by the electromagnetic radiation. While the unidirectional motion of the electromagnetic wave and the relativistic electrons provides for longer overall capture of the particles by the field, protracted drift of an electron across the accelerating part of the monocyclic THz pulse may be achieved by calibrating the additional magnetic field. Simulations demonstrate that, under optimized parameters, the eventual electron energy gains may reach the magnitudes ranging from several to tens of electron rest energies depending on the injection energy. The suggested scheme of post-acceleration of relativistic electrons by combinations of comparatively moderate-intensity THz electromagnetic pulses and constant magnetic fields may thereby provide a viable alternative to the conventional laser accelerator designs employing extremely powerful laser systems that generate relativistically intense electromagnetic envelopes.