I stayed in Lone Pine July 17-19 and did a couple of day hikes in the area. On Day 2, I decided to hike to Kearsarge Pass, with the possibility of trying Mt. Gould. Unfortunately some weather rolled in as I was hanging out at the pass, so I decided to save Gould for another day.
Here are some hightlights. Clicking on any of the photos will bring you to my full trip report and gallery (~60 photos):
So, I guess the last reliable water is at Flower Lake en route to Kearsarge Pass? Just looking at my map, it doesn't look as though the trail goes close to Heart Lake.