I'm not sure I can help you with "timing", but I can say that we were able to obtain two Whitney permits on July 3 (we wanted to hike up to Trail Camp for acclimatization) and another 2 on July 5 (we summitted). Although you could make an argument that the late season snow caused a lot of cancelations, we were still able to get 2 permits 2 times on what has to be a very busy time for the system. We just monitored availability, in both cases we got them the day before. Good luck!
