A coupled-perturbed Kohn-Sham treatment for the calculation of hyperfine tensors has been implemented into the MAG-ReSpect program. It treats spin-orbit contributions to hyperfine tensors by a combination of accurate and efficient approximations to the one- and two-electron spin-orbit Hamiltonians: (a) by the all-electron atomic mean-field approximation, and (b) by spin-orbit pseudopotentials. In contrast to a previous implementation, the code allows the use of hybrid functionals and lifts restrictions in the orbital and auxiliary basis sets that may be employed. Validation calculations have been performed on various transition metal complexes, as well as on a series of small diatomic molecules. In the case of a series of copper(II) complexes, the spin-orbit contributions are large, and their inclusion is essential to achieve agreement with experiment. Calculations with spin-orbit pseudopotentials allow the efficient simultaneous introduction of scalar relativistic and spin-orbit effects in the case of light nuclei in the neighborhood of heavy atoms.