The very-low-temperature thermal conductivity (κ) is studied for BaCo2V2O8, a quasi-onedimensional Ising-like antiferromagnet exhibiting an unusual magnetic-field-induced order-todisorder transition. The nearly isotropic transport in the longitudinal field indicates that the magnetic excitations scatter phonons rather than conduct heat. The field dependence of κ shows a sudden drop at ∼ 4 T, where the system undergoes the transition from the Néel order to the incommensurate state. Another dip at lower field of ∼ 3 T indicates an unknown magnetic transition, which is likely due to the spin-flop transition. Moreover, the κ(H) in the transverse field shows a very deep valley-like feature, which moves slightly to higher field and becomes sharper upon lowering the temperature. This indicates a magnetic transition induced by the transverse field, which however is not predicted by the present theories for this low-dimensional spin system.