I don't think there's a way to handle this situation that is perfectly risk-free, but I do think that arranging alternate interview days (rather than rescheduling at other programs) is the best way to go.
Depending on the program, you may have less contact with faculty and students at alternate interviews, because you won't be there for department-wide events on the regular interview days. However, not all programs have extensive interview weekend programming, and those that do (like mine) may make efforts to have you meet as many people as possible even on alternate days. On the plus side, when you go on an alternate day, you're more likely to be the only person interviewing that day, so you might be able to make more of an impression.
Conversely, I think that trying to reschedule interviews that are already set is a riskier choice, because you're letting those sites know that you're less interested in them.