We study the low-temperature heat transport, as well as the magnetization and the specific heat, of TmMnO_3 single crystals to probe the transitions of magnetic structure induced by magnetic field. It is found that the low-T thermal conductivity (kappa) shows strong magnetic-field dependence and the overall behaviors can be understood in the scenario of magnetic scattering on phonons. In addition, a strong dip-like feature shows up in kappa(H) isotherms at 3.5--4 T for H parallel c, which is related to a known spin re-orientation of Mn^{3+} moments. The absence of this phenomenon for H parallel a indicates that the magnetic-structure transition of TmMnO_3 cannot be driven by the in-plane field. In comparison, the magnetothermal conductivity of TmMnO_3 is much larger than that of YMnO_3 but smaller than that of HoMnO_3, indicating that the magnetisms of rare-earth ions are playing the key role in the spin-phonon coupling of the hexagonal manganites.
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