My brother, a friend of his and I did this two years ago last October. That I think, like the Grand Canyon RTR hike, is the best month to go. Won't be any snow and it won't be that hot either. As mentioned above, start predawn, preferably by 5:00 am, because even in October it will be hot in Palm Springs in October once the sun comes out. You want to be at least 2-3 thousand feet of vert into the hike before that happens. Otherwise it makes for a miserable start to the hike and most important, you'll need more of the valuable water you're hauling along.

Very advisable to bring along a GPX track of the route because it's easy to lose track of the trail. We did the complete hike, from museum to peak and back to museum. Going down at night was a challenge navigational-wise and 100% concentration was required from the point man. My hiking partners have sworn they'll never do it again (at least going back down to the museum) but I thought it was pretty cool seeing the Palm Springs lights become more prominent as we descended and I have good memories of it. This hike CTCTC compares with RTRTR, but I think the latter is harder. The main difference is that RTRTR is longer (43-48 miles, depending on which route you choose) and that CTCTC is all UP then all DOWN. But many people are afraid of the stress on their legs of 10k of continuous downhill and I don't blame them. Fortunately it wasn't an issue for me like it was for my hiking partners.