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If you give them addressed stamped envelopes (personally is better), write nice notes to their secretaries if they have them, give them all the information needed and envelopes all at once so they only have to work on it once, and were a good student, then the marginal cost of another letter low enough that they shouldn't mind another 10-15.
Tell them each the situation and that you would like to apply to X institutions, and ask if they would write another letter. You will, of course, need to tell them why you are changing, etc, but I think they will all consent.
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