Kudos to Otherhand and Adam for continuing the search!
Report's up, and looking over the search tracks relative to what's been done, on my end I think the next logical place to look is the foot of the mountains on the east side of Quail Wash in the same valley. There are spots that apparently haven't been looked at yet, and if Bill was walking out from Quail Wash, that's the most logical place for him to have gone for shade, as it's closest to the wash and doesn't require the detouring that the area I searched last week would have. Might be fun to enter that region up and over the hills from the JTNP boundary (which also isn't searched territory).
That's exactly where I was going to suggest to look next in pursuing the Quail Wash theory.
As you've probably already thought through, if Bill is travelling north through the wash after pinging the tower, then it is still early morning (7am) and the sun will be rising in the east. As the sun continues to rise, to get shade, Bill would have to shelter on the western side of something. That would be easiest to accomplish by moving east off the trail and sheltering on the west side of a rock pile or cliff face. However, a position that is in the shade at 9am would not be in the shade at 1pm, when the sun is overhead, or 6pm, when the sun is in the west. Thus to remain in the shade, which has probably become an issue of survival, Bill might have changed positions one or more times, while not leaving a small area. Late in the day, he would be sheltering on the east side of something, or under something, to stay out of the sun in the west. As a result, he might have ended up in an unexpected location in one of the side canyons on the east side of the wash, and the location might not be visible to people moving north-south in the wash.
The timing of the ping has always given me heartburn. My first guess is that Bill was probably not moving after Thursday night, and ping is tied to some other scenario, like animal activity. If Bill did cause the ping from Quail Wash, my guess is that he climbed either to Johnny Lang Mine or Quail Mountain, suffered a fall and received a concussion, and was immobile from Thursday to Saturday night. I had a concussion when I was young and the 2 primary effects were that I was stuck in bed for 2 or 3 days before I could walk around, and memory loss, including loss of memory from before I received the concussion.
So assume Bill is concussed Thursday at one of those two locations. While he knows that he is in Joshua Tree, he might not be able to remember his actual location in the park, the close proximity of the park road, or the location of his car. He doesn't feel able to move until Saturday evening. What does he do? Saturday evening, he heads for the lights on the northern park boundary. He heads down Johnny Lang Canyon in the dark (the sketchiest part of this theory), passes through the ping area at 6:40am, and proceeds up the wash. At some point, the combined effects of the concussion and dehydration force him to find shelter on the east side of the wash.