If you are looking for record breakers or bragging rights, mt San Gorgonio via Vivian trail is a knee punisher about 16 to 20 miles, it's the highest mountain in SOUTHERN calif. Of course mt Whitney being the highest in calif. It's nice, it's green, it's very crowded, it requires a permit, and you will prob get altitude sickness. It's best for 2 days to enjoy the forest and acclimate to altitude. Look it up on localhikes.com
Mt baldy aka San Antonio peak is a requirement for socal hikers.
San jacinto peak is very scenic, permit required.
The tourists like the Griffith park mt Hollywood (sign) area.
Mt Lowe has some historic stuff, and abandoned villa with a train on the summit
Death valley is closer to SEKI, but in June, your car might overheat.
Anza borrego desert, and Joshua tree (yes the U2 song) is pretty cool, kinda hot for June, look up Carrey's Castle.
Malibu has some nice areas point mugu, sycamore canyon.
Santa Monica mountains.
Depends if you looking for scenic, miles, records, peaks, easy going. I'm guessing you want scenery you have not seen on the east coast.