I did the section from mile 5 of the JMT to Red's Meadow last year, and I also found the planning very confusing intially. Having a map in front of you helps tremendously. I started from near Tuolumne, about nine miles down the road at Sunrise. You can hike it from Tuolumne - some road miles and some along Tenaya Lake, I think, but I hitched. Spent the night before in the Tuolumne backpacker's camp, which you can do if you have a permit for the next day. The Sunrise permit is easier to get than any from the valley (according to YNP staff), and you can include Half Dome if you want (I didn't). You loop down to the JMT via Cloud's Rest and join it (the JMT) at mile 5. (The turnoff to Half Dome is at mile 4.5, and you could do it on the second day out). From there, you just hike the JMT, missing only the first 4.5 to 5 miles.
It is about 11 or so miles (don't have my maps) from Sunrise to the JMT via Cloud's Rest, but you don't have to go over Cloud's Rest - there's a trail that skirts it, I think. I camped up top on my first night out and it was quite nice, but that might not be a good idea in Jul/Aug (I was there in late Sept).
Anyway, I thought I'd let you know that Sunrise is an option if you can't get any of the Happy Isles options, or if, like me, you can but don't fancy the long slog out of the valley.
Also, there's a JMT yahoo group that is extremely helpful with things like this (and with everything about the JMT, actually).