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In this work, we explore the possibility of enhancing a spin current under a thermal switch, i.e., connecting the central transport region to two leads in individual thermal equilibrium abruptly. Using the nonequilibrium Green's function method for the transient spin current, we obtain a closed-form solution, which is applicable in the whole nonlinear quantum transport regime with a significant reduction of computational complexity. Furthermore, we perform a model calculation on a single-level quantum dot with Lorentzian linewidth. It shows that the transient spin current may vary spatially, causing spin accumulation or depletion in the central region. Moreover, general enhancement of the spin current in the transient regime is observed. In particular, the in-plane components of the transient spin current may increase by 2-3 orders of magnitude compared to the steady-state thermoelectric spin current under a temperature difference of 30 K. Our research demonstrates that ultrafast enhancement of spin currents can be effectively achieved by thermal switches., Comment: Submitted to J. Phys. D: Appl. Phys. (http://iopscience.iop.org/article/10.1088/1361-6463/aac7ca/meta)