I stayed at Hawk twice on a loop through Marin County five years ago:
https://backpackinglight.com/forums/topic/90733/
Days after a big rainstorm, I found a large puddle of muddy water along the approach trail/road a couple hundred yards below camp. Didn’t look inviting and I was already carrying a bunch of water, so didn’t try to treat and drink. Picked up water at the Marin Headlands Rodeo Lagoon visitor center about 3.7 miles before Hawk on the return leg of the loop.
Passed the horse stables, specifically looking around for water. Didn’t see anything, but I wasn’t bold enough to walk in and ask, either.
At the coastal end of Tennessee Valley is a pond that almost always has water. The creek feeding it is seasonal. https://www.google.com/maps/@37.8436646,-122.5504088,581m/data=!3m1!1e3
HTH.
— Rex