Summers here in SoCal the ground water is too hot to adequately chill wort. I generally get the wort down to about 80F and then bring it down to pitching temperatures in a refrigerator. I also have a double chill coil available in an ice bath for the final few degrees.
In the past, I’ve pre-boiled additional store bought water on the stove, then chilled down to 34F in one of my lagering refrigerators. You can safely top up with this. You probably should use RO or distilled water, unless you are really dialed in to the mineral content of whatever water you are adding. I’ve also topped up with straight distilled water right off the shelf without issues, but you should boil it again anyway just to be on the safe side. Make sure to spray the jugs with Star San solution and wipe them down before transferring.
I wouldn’t use SoCal tap water for anything except showering. I can seasonally detect organic odors in the water, I’m not convinced it is safe to add to wort either unboiled or ice cubes. Plus Chloramine. Eck!