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We study the heat transport of S=1/2 chain compound CuCl2 · 2((CH3)2SO) along the b axis (vertical to the chain direction) at very low temperatures. The zero-field thermal conductivity (j) shows a distinct kink at about 0.9 K, which is related to the long-range antiferromagnetic (AF) transition. With applying magnetic field along the c axis, j(H) curves also show distinct changes at the phase boundaries between the AF and the high-field disordered states. These results indicate a strong spin-phonon interaction and the magnetic excitations play a role in the b-axis heat transport as phonon scatterers. [ABSTRACT FROM AUTHOR]