Oops, just re-read and I see the early June start --- a little awkward, a late start for a NOBO, and perhaps a little early (?) for a PCT SOBO.
In fact, PCT thru-hikers (NOBOs) tend to be coming through the Sierras (JMT) in June, but jumping in at that point would be a bit weird --- they would be in fantastic shape, while you would just be getting started. Perhaps just jump on trail in Mojave or perhaps Agua Dulce, which would give you a feel for some aspects of SoCal, and give you a chance to get your hiker legs before walking into the Sierras.
Obviously there are a ton of other things a person could do with three months (!), so this idea depends very much on how interested you are (or are not) in the sort of "thru-hiker experience".