I try to do our my booking on CO.com, as we need the miles to stay elite.

We commonly fly CLE-SAN to spoil our granddaughter. Since we've seen about all the museums in San Diego, I thought flying CLE-LAX-SAN, with a stopover in LAX to check out some of their museums would be interesting. I know CO flies this route, with the LAX-SAN leg on, I think, American Eagle, as when I search on CO.com for round trips CLE to SAN, sometimes a few flights including LAX show up. In fact, when I search using Continental Airlines Timetable, quite a few of the American Eagle flights are listed. However, when I search on CO.com for a multiple destination ticket with a stopover in Los Angeles, it consistently reports it cannot find such flights.

What is the best way of booking such a flight, while at the same time using CO.com so we get the maximum miles?