I went with Dana on my trip in 2012. I think it was like June 30-July 6 or so. We had a great week. We caught good numbers of both chinook and steelhead. We fished the lower river only for that week. I am not sure which lodges, if any, fish the upper river early,you would have to check. If I was going to fish the upper river (or have the option to fish lower and upper), I would likely target late July or August and target steelhead. Good dry fly opportunity as well then. If you want both, then I think doing a lodge on the lower river in the first week of July might be your best bet. If your primary taget is chinook then maybe the last week of June and you still might get an early steelhead.
Good luck. I enjoyed my trip immensely. The lodges usually work together and each alter their program (fishing times) a little so as to better share the river (or at least they did when I was there). If I go to the Deam again, I am going to try the August program on the upper river.