We study the improvement of non-permutation over permutation schedules in flow shops when minimizing the total completion time. We solve both problems by a two-phase heuristic. The first phase uses an iterated local search to find a good permutation schedule. The second phase explores non-permutation schedules using an effective insertion neighborhood, that permits to anticipate or delay a job when passing from one machine to the next. In computational experiments we show that both phases yield state-of-the-art results. We find that allowing non-permutation schedules can reduce the total completion considerably with a moderate extra effort, and without increasing the buffer size needed during processing. We conclude that nonpermutation schedules can be viable alternative to permutation schedules in flow shops.