The paper outlines the hybrid framework of spin hydrodynamics, combining classical kinetic theory with the Israel–Stewart method of introducing dissipation. The local equilibrium expressions for the baryon current, the energy–momentum tensor, and the spin tensor of particles with spin 1/2 following the Fermi–Dirac statistics are obtained and compared with the earlier derived versions where the Boltzmann approximation was used. The expressions in the two cases are found to have the same form, but the coefficients are shown to be governed by different functions. The relative differences between the tensor coefficients in the Fermi–Dirac and Boltzmann cases are found to grow exponentially with the baryon chemical potential. In the proposed formalism, nonequilibrium processes are studied including mathematically possible dissipative corrections. Standard conservation laws are applied, and the condition of positive entropy production is shown to allow for the transfer between the spin and orbital parts of angular momentum.