Congratulations on your nice score. It should be an easy grow for you in Irvine. More shade will translate to deep green and large leaves at the expense of flower production. The foliage cover of your plant is a little on the sparse side, but in time it will fill up and become a very dense tree. Before that happens, you might need to give it some sun-scorching protection when it gets really hot (90F plus).

M. X alba needs acidic well-drained soil with high organic content. So mix lots of pumice/perlite/lava rock/sulfur/compost/peat moss into your existing soil. A great fertilizer recipe I got from a commercial grower is to mix 4 tablespoons of fish emulsion and 1 tablespoon of water soluble 20/20/20 such as Miracle Grow in 5 gallons of water. Use one or two cups of this every time you water during the growing season (for your location, it is probably from April to November). Skip once a month and let regular water flush out accumulated salt. Try it and you will be amazed at the amount of new growth and flowers. Fish Emulsion does smell fishy but that usually dissipates in a day. What I do now is to dilute this dosage by half and use it even more liberally.

One other thing to consider is to keep heading back your M. X alba so that it stays shrubby. Its natural tendency is to grow into a 20-30' tall tree in a few years, which will be good for flower production but you will have a hard time harvesting them.

Do not over-fertilize with nitrogen or you might get gigantic leaves and weak branching. See how the leaves on my plant are weighing down the branches and causing them to grow downwards and gnarled up. The new branches on the left are on that verge too. They are growing horizontally at the moment unlike the upwards way of yours.

My tree looked worse earlier this year but I trimmed most of the ugly branches (but you can still see some near the bottom right). The old leader also died from sun damage and a new leader popped up from near the base (left side in pic). That's ~4ft+ of growth in one season!

And I do not normally have this out in the open, it usually stays under the tarp canopy behind it. Otherwise the leaves will get sun damage. This also protects it somewhat from those vicious Santa Ana winds.

I live about 30 miles inland from you so it is much more arid but perhaps yours could handle full sun with the higher humidity near the coast.