The spin-Seebeck effect together with a high spin thermoelectric conversion efficiency has been regarded as one of the core topics in spin caloritronics. In this work, we propose a spin caloritronic device constructed on hydrogen-terminated sawtooth graphene-like nanoribbons periodically embedded with four- and eight-membered rings to investigate the thermal spin currents and thermoelectric properties by using density functional theory combined with the non-equilibrium Green's function method. Our theoretical results show that spin-Seebeck currents are induced by the temperature gradient between two leads due to two isolated spin-up and spin-down transport channels above or below the Fermi level. Besides, the embedded four- and eight-membered rings break the mirror symmetry of graphene-like nanoribbons and increase the phonon scattering to lower the lattice conductivity, contributing to the enhancement of the spin figure of merit. Moreover, the increasing width of the nanoribbons can effectively enhance the spin-Seebeck currents and reduce their threshold temperatures to improve the device performances. These systematic investigations not only give us an in-depth understanding into the realistic spin caloritronic device applications of graphene-like nanoribbons, but also help us to choose feasible routes to improve the spin-Seebeck effect with a high spin figure of merit in nanostructures.