Are you starting from the valley floor or from the backcountry?
There is good spring right on the trail soon after the JMT junction and less than 2 miles from the summit. If you are coming down from the back country, there is also water on the JMT near the junction to the Cloud’s Rest trail.
If you are hiking from the Valley, I would recommend getting up as early as possible to start the hike (4am). Even better is camping overnight in the backcountry (you’ll need a permit) the night before. In either case, your goal is to get to the cables at the base of Half-Dome as early as possible. It gets very crowded later in the morning (as in you might have to wait in line to get on the cables). Climbing the cables is a bit of a physical and mental challenge and I think it is easier with fewer people around.
A 1/2 liter plastic water bottle doesn’t seem like enough to me. It is 17+ miles round trip from the valley and over 5,000 feet of elevation change. Be prepared! I’d also recommend a pair of gloves to use on the cables. There is a pile of gloves at the start of the cables from previous hikers but you might feel better using a pair that you brought along.
If you are hiking down from Tuolumne, I’d recommend going over Cloud’s Rest for a fantastic 360 degree view of Yosemite.
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