  We present a new derivation of Israel-Stewart-like relativistic second-order
dissipative spin hydrodynamic equations using the entropy current approach. In
our analysis, we consider a general energy-momentum tensor with symmetric and
anti-symmetric parts. Moreover, the spin tensor, which is not separately
conserved, has a simple phenomenological form that is antisymmetric only in the
last two indices. Apart from the evolution equations for energy density, fluid
flow, and spin density, we also find relaxation-type dynamical equations for
various dissipative currents. The latter are consistently derived within the
second-order theory as gradient corrections to the energy-momentum and spin
tensors. We argue that this approach correctly reproduces the corresponding
Navier-Stokes limit of spin hydrodynamic equations. Throughout our analysis,
the spin chemical potential is considered a $\mathcal{O}(\partial)$ quantity in
the hydrodynamic gradient expansion and reduces to thermal vorticity in the
global equilibrium. New coefficients appearing in the generalized spin
hydrodynamic equations are undetermined and can only be evaluated within a
proper underlying microscopic theory of a given system.
