Depending on how much water you drink in a day, you should be fine for the whole day. If you need to refill or top off, I would just buy a water bottle and pour it into the reservoir. The water from the park fountains is nasty tasting.
But then again, I'm not used to how California water tastes.
First Aid on Main Street has filtered water. Rancho Zocalo has a spigot. Most places will give you a glass for water and a lot of the soda machines filter the water. Though I don't usually have to fill up, my CamelBak has a pretty good sized bladder.
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