You must fill up at trail camp lake as there's no other source of water except snow near the top and some dinky little trickles of water here and there. But there's always a chance of bacteria in the snow so I wouldn't go for it unless you're desperate, especially if the snow is reddish. I had water issues myself (among many other issues that day). We had a lagging team member we had to wait for and that made us use up too much of our supply. Just make sure you ration. I brought 5L from trail camp and it wasn't enough. I should have rationed better or maybe tried bringing 6L, but that's heavy. My body is not very efficient with water however, and some of you might actually be so it depends.
Hannah, many people will try to fill up at switchback #23 above Trail Camp. I would hope that water is still running. It doesn't usually dry up until mid-August.
Most people get by taking 3 liters of water from Trail Camp (or that switchback). Drink one on the way up to Trail Crest, the next on the way to the summit, and #3 on the way down. Bob R (now retired from hiking) would always supplement his water by refilling or topping up his water with snow at Trail Crest and at the clean patch near the summit.
Drank tons of fluids and ate well. The only problem was having to get out of the warm sleeping bag 3 times during the night.
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I picked up a solution to that from an x-c skier at Pear Lk lodge in Sequoia years ago. Now I carry and use a 20 oz Gatorade bottle with an "X" on the cap. If you think you might need to use it more than once, dig a pit within arms reach of your bed.
For women, it's a little more difficult, but the "
Freshette" from REI can help a lot.