The morphological transitions during directional quenching-induced spinodal decomposition in binary mixtures are investigated by computer simulation. By setting the quenching front between the stable and unstable phases, and shifting the front with a constant velocity, the evolution of the domain morphologies is examined numerically on the basis of the time-dependent Ginzburg-Landau (TDGL) equation. Three different types of morphologies are found for the critical quenching. One is irregular morphology (IM), which is essentially equivalent to that produced by homogeneous quenching. The other two are regular, representing the characteristics of the directional quenching process. One is regular lamellar morphology (RLM) and the other is regular column morphology (RCM). By varying the shifting velocity of the cooling front, two morphological transition velocities, v(a) from IM to RLM, and v(i) from RLM to RCM, are observed. In contrast to that, for the case of off-critical quenching, a new transition velocity v(b) from RCM back to RLM can be found if the cooling front is further shifted slower. This characteristic morphological transition is attributed to the surface enrichment effect appearing in the nonequal volume fraction system, which competes with linear instability triggered by initial thermal fluctuation in the early stage of spinodal decomposition. Detailed studies reveal that RLM can be easily formed and thus the region of RCM is reduced when the surface enrichment effect is stronger. On the other hand, RCM will be preferred if the initial thermal fluctuation is stronger. The quantitative relation between lamella width and shifting velocity of the cooling front is also presented.