It's a long ways from Hamilton to the Junction. The store there had water last time I was there, inside in the bar. But you should buy something too.
You can also get water from the fire station on the other side of the road but it is nasty tasting.
It's a long ways from the Junction to Livermore. We usually stop for food and water at the Safeway on Concannon.
Last time I did this we found water at Sunol park- the first set of buildings on the left after you go past the entrance station.