Hi there! I just graduated in May with my BSN took my NCLEX July 28th, applied to St. Joe's for a Clin Nurse I Labor and Delivery position and accepted it :) I've been working there for almost a month and love it!
The whole St. Joseph's healthcare system actually has a pretty developed a residency program since they merged with Providence Heath last year in 2016 and it's much more developed than I had thought! At St. Joe's, St. Jude's, and Mission you are required to have your RN license to even apply though. I believe this is true at Hoag and at UCI as well (UCI might accept an interim permit, which you can apply for when you go to apply for you license a little before graduation).
The peds hospitals (CHOC, CHLA, Rady's) and I think UCLA accepted applications without a license.
You also wanna start checking out what other certifications you might need (ACLS, PALS, NRP, fetal monitoring, ect.) that might put you ahead of other applications.
Also, when in doubt, every hospital has a designated nurse recruiter that you can 1) call the general HR # on the website and 2) just ask to speak with he recruiter to ask clarifying questions like "do I need my license to apply" and "when does the program officially start" and "how many new grads are you expected to hire" ect.