LGB Controller described a unsafe condition when SCT failed to provide timely coordination regarding a Runway 30 arrival after other aircraft were issued straight in approvals.

Narrative

I received a call from Aircraft X; VFR; requesting arrival to Runway 30. Using the RACD; I scoped out approximately 20 miles to see if there were any aircraft on approach before approving the operation. I wanted to ensure I did not disrupt the Runway 30 arrival sequence. There was no one in the vicinity tagged IFR to LGB. I then instructed Aircraft X to follow Aircraft Y; who was sequenced number 2 by SCT for the ILS approach to Runway 30; preceded by Aircraft Z. I did not have either IFR aircraft on my frequency at the time; but Aircraft X was able to see and follow Aircraft Y. I received a call shortly after near the FAF from Aircraft Y advising me of his location and intentions. I cleared him to land; number 2; following traffic ahead; who had not called me at this time and was already inside my airspace. I reached out to him and he replied. I then cleared Aircraft Z to land; number 1. It appeared that Aircraft X was following closer now; so I checked back with him ensuring he had his aircraft in sight. He answered in the affirmative. When Aircraft X was on approximately a 3 mile final; I received a call from Aircraft A. He stated that SCT had just handed him off and he was over the Pyramid. He was tagged up at this time landing LGB right over the top and overtaking Aircraft X. I quickly called out traffic to Aircraft A; who stated he had passed a helicopter. I asked Aircraft X of his position and he advised he had close in traffic in sight. I then turned Aircraft X off the final approach for Runway 30 for traffic. I was unaware of Aircraft A's presence prior to his call as he was tagged very close to the FAF and not properly handed off or pointed out prior to that point. As Aircraft A overflew Aircraft X; I had been scanning my runways and out the windows for other traffic. Aircraft A was nearly twice the speed of the 2 aircraft he was sequenced to follow by SCT. It seemed that he just came out of nowhere when I scanned back to my RACD to locate him and other traffic. When Aircraft A was on final; I also needed to have him execute S-turns for runway spacing due to his speed in trail of Aircraft Y. Had I been advised of Aircraft A at the proper time I could have taken Aircraft X off of his approach to Runway 30 in a timely manner as to avoid Aircraft A; who was sequenced by SCT. [I] recommend ensuring [that] SCT is pointing out Runway 30 inbounds prior to 3 miles outside the FAF to LGB regardless of IFR or VFR status. Provide adequate spacing between successive VFR/IFR arrivals to LGB to accommodate runway spacing. We are often given numerous aircraft sequentially too close to provide adequate runway spacing and this is a prime example. Ensure that aircraft are verbally handed off to LGB prior to the FAF to ensure we can provide separation between them and other aircraft in our airspace.