I don’t think anything is going to look great out there. What is the back situation? That probably would be a better lounge spot.

Thanks that's kind of what I thought too, any ideas on what else we can do to add shade? Potted plants, different furniture?

You can go with attached or detached pergola.

Some rough visual here

or

https://www.lowes.com/pd/DuraTrel-105-in-W-x-105-in-L-x-96-in-White-Plastic-Freestanding-Pergola/1000786150

It looks like the house faces west, so you need shade from afternoon sun, both high and low. I see a young tree near the chairs on the left (can't tell what kind); as that grows, it will provide some overhead shade. A couple of small trees (fruitless olive?) in front near the sidewalk would (a) block the lower late-afternoon sun, (b) give a sense of privacy to the lawn, and (c) increase the sense of distance from street to house.

I would also want some privacy from the house on the left. It looks like the concrete patio where the chairs are extends to the property line, so perhaps planter boxes set on the concrete with bamboo for feathery screening? Something like these?

An arc of native shrubs (ceanothus? choose a smaller cultivar) left of the lawn would add a sense of separation/enclosure, and a few low plantings along the walkways would add interest to those journeys. Shown are silver lupine (Lupinus albifrons) and foothill penstemon (Penstemon heterophyllus 'Margarita'), both native, plus Mexican feather grass (Nassella tenuissima), which is not native to California and does seed; check locally before using and substitute as needed. Does the brick make planters on each side of the bay window? If so, plants there would add additional cooling greenery.

Some ideas, anyway . . .

We live in a very sunny climate and have four cantilever umbrellas on wheeled bases in our yard. We find them extremely practical. With a pergola, morning sun will throw the shade twenty feet to the west and evening sun will throw the shade twenty feet to the east of the structure. The shade also moves from north of the structure to south as the days get longer. Wheeled cantilever umbrellas can be rolled from one side of the seating area to the other (east side to west side), can be tilted from flat overhead to almost straight up and down when the sun is low, and ours also rotate 360 degrees. We have the brand “Purple Leaf” and are very pleased with them, but this screenshot gives you the idea: