In the literature of two-sided matching, each agent is assumed to have a complete preference. In practice, however, each agent initially has only partial information and needs to refine it by costly actions (interviews). For one-to-one matching with partial information, the student-proposing Lazy Gale-Shapley policy (LGS) minimizes the number of interviews when colleges have identical partial preferences. This paper extends LGS to a significantly more practical many-to-one setting, in which a college can accept multiple students up to its quota. Our extended LGS uses a student hierarchy and its performance (in terms of the required number of interviews) depends on the choice of this hierarchy. We prove that when colleges’ partial preferences satisfy a condition called compatibility, we can obtain an optimal hierarchy that minimizes the number of interviews in polynomial-time. Furthermore, we propose a heuristic method to obtain a reasonable hierarchy when compatibility fails. We experimentally confirm that compatibility is actually much weaker than being identical, i.e., when the partial preferences of each college are obtained by adding noise to an ideal true preference, our requirement is much more robust against such noise. We also experimentally confirm that our heuristic method obtains a reasonable hierarchy to reduce the number of required interviews.