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A Better Way to Match Exchange Students and Schools?

Students in the Maastricht University exchange program provide an ordered list of their six most-preferred schools. Despite this, because their preferences matched many other students’ preferences and/or they were far back in the queues, 47 of the 357 participating students have not yet been placed.
Is there any way to do a better job of matching? Does the path-breaking work of Alvin Roth (more here) give any guidelines to improving the efficiency of this complicated matching scheme? One possibility is a two-stage process, asking students who are unplaced after Round 1 to list their preferences over the remaining available list of unfilled places. One could even iterate additional rounds until all students are placed. Both students and those foreign universities that are not attracting enough Maastricht students would be better off. But can one do better than this simple addition to the matching process?