Can you add to the top of the existing fence? I’ve also had potato vine in a couple houses and it takes a life of its own -grows really fast against walls/fence, but you’d have to trim it back in order to not have it encroach. We couldn’t hear or see the neighbors next door to us when we lived in LB with that vine growing between us. I’m attaching a pic from the backyard...the blue line is where I think the top of the wall/fence was.
You won’t have something immediately, but by next summer you will have quite a lot of coverage/privacy.

Plant trees along the fence. I’m not sure what varieties work on the left coast, here I would plant crepe myrtles close to the fence. Yes you give up two feet of lawn but you quickly get a much larger canopy above the fence. Have you been to a major landscape supply in your area to see what they have? Often they have options that are quite large, most large suppliers will also plant for you for a fee (when they get to a certain size they get very hard to transport and wrangle)

Cultivars of Polyscias guilfoylei are a common tall skinny hedge in Hawaii. You might be able to find it in LA. Called panax commonly and mostly a house plant on the mainland.

I always love the look of Italian Cypress.

https://paradisenursery.com/product/italian-cypress/

IMO if you want instant buy big trees

Podocarpus makes an excellent privacy screen for SoCal and grows pretty darn fast, all things considered :-) If you and your neighbor invest in larger specimens to begin with, you will both have plenty of privacy in less than 3 years.

Another alternative for fast yet narrow privacy screening is any type of clumping bamboo - Bambusa, Fargesia or Borinda species.

Another vote for clumping bamboo. Link to nurseries in LA that specialize in bamboo.

https://www.google.com/search?rlz=1C1CHBF_enUS854US855&sxsrf=ALeKk01IgCU07v8UMzP92QVNKeCiQgqqgw%3A1588172118338&ei=VpWpXtyQFMO0swX9wIXYAg&q=bamboo+specialty+plant+nursery+los+angeles&oq=bamboo+specialty+plant+nursery+los&gs_lcp=CgZwc3ktYWIQARgAMgUIIRCgATIFCCEQoAE6BAgAEEc6BwgjEOoCECc6BQgAEJECOgIIADoFCAAQgwE6BAgjECc6BAgAEEM6BAgAEAo6BggAEAoQQzoECAAQDToGCAAQFhAeOgYIABANEB46CAgAEAgQDRAeOgQIABAeOgYIABAIEB46CAghEBYQHRAeOgUIIRCrAlCo3j5Y3ZNAYPWkQGgBcAJ4AIABkgGIAYAdkgEFMTkuMTeYAQCgAQGqAQdnd3Mtd2l6sAEK&sclient=psy-ab

The fastest-growing hedge in Florida is Clusia Rosea or the Pitch Apple plant. Purchase in 3 or 7-gallon containers. It will grow faster than podocarpus and has no issue with pests. They are the go-to hedge plant since the cities in SWFLA have prohibited Ficus Hedges because of their invasive qualities. Drought tolerant

• The recommended spacing is 1 foot apart for the 3g clusias, 1.5 apart for the 7g clusias, 2.5 feet apart for the 15g clusias and 4 feet apart for the 25g clusias. Once in the trench and they are looking good remember that clusias love the sandy soil and this will not be a problem.

BTW I see you have two Queen Palms. They are the cheapest palms to purchase and the most expensive to maintain. Their fronds to not fall off automatically so you have to hire someone to prune them when they grow too tall. They are prone to magnesium deficiencies so they need fertilizer to stay strong. They are a bad investment.

Use Bromeliads for Color

I would only add that what grows well in Florida is not necessarily a great choice for SoCal :-) They are very different climates!!

Thank you everyone for all this great advice! I think building up the fence and then adding trees that don't need a lot of ground space is the way to go. Just need to figure out the right trees for our SoCal zone and that won't be problematic for our tiny little grassy lawn.

Sorry, for mistaking this for FLA

Just an aside. The two last pictures of British gardens show established pleached trees. They're very expensive to buy. They need a lot of maintenance, on both sides. And they're not olives.

Is there space in the corner to create a sitting area and install one of those cool fabric overhangs ie a shade sail? I think a pop of color and seating would be nice there. We masked a neighbors house with an umbrella table (angled the umbrella at my sisters and it worked far better than landscaping - if done right it would be lovely and wouldnt take allot of space with added benefit of having more space for seating

Another aside: my (New England) town limits fences to 6' high, tho perhaps it's a local issue & not in CA?

@deb s yes actually was thinking of doing shade sails! Love this idea. @blazegirl I think we just need to pull a permit to put in a taller fence, I'll ask my contractor. Not back to researching trees :-)