  We investigate the spin Seebeck coefficient $S_s$ in the square lattice
Hubbard model at high temperatures of relevance to cold-atom measurements. We
solve the model with the finite-temperature Lanczos and with the dynamical
mean-field theory methods and find they give similar results in the considered
regime. $S_s$ exceeds the atomic 'Heikes' estimates and the Kelvin entropic
estimates drastically. We analyze the behavior in terms of a mapping onto the
problem of a doped attractive model and derive an approximate expression that
allows relating the enhancement of $S_s$ to distinct scattering of the
spin-majority and the spin-minority excitations. Our analysis reveals the
limitations of entropic interpretations of Seebeck coefficient even in the
high-temperature regime. Large values of $S_s$ could be observed on optical
lattices. We also calculate the full diffusion matrix. We quantify the
spin-thermal diffusion, that is, the extent of the mixing between the spin and
the thermal diffusion and discuss the results in the context of recent
measurements of the spin-diffusion constant in cold atoms.
