A pergola is one way to go. Some come with a retractable canopy for full shade.

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This one is from Wayfair:

Budget sun shade sail. This is also from Wayfair:

Show a picture of the entire yard. We need to see where the patio is in relation to the house.

The best thing we did was get rid of the umbrella and attach an awning to the house. We planted a tree years ago but it took a long time to offer shade and now all we do is sweep the samara off the deck and patio.

We have our pool on the very sunny side of the house and big trees in the other sunny spot for seating. You could do a pergola but we need to see the whole space .

We had a pergola installed above our existing deck last fall. It wasn’t cheap but it’s handsome and very effective for shade. It is styled like a neighbor’s who has a free-standing one at the edge of his yard overlooking the golf course.

It's very easy to DIY a tarp awning. Install 4 tall posts around the perimeter of the patio. Use bungee cords in the grommets to secure the tarp to the posts overhead with post hooks.

You can use sailcloth that lets some light through or just solid cloth.

Tons of variations.

Still waiting for pics from OP

The OP

We had a similar situation--our backyard faces west (not optimal, I know, but we have farm views), and there were no trees on the property when we built the house. The builder assured us that it wouldn't be a problem because of the one-foot overhang on the second floor. Of course that's laughable, and I said that wouldn't be enough to block the PM sun, but no one listened to a young mother at that time. Of course, once we moved in, we found that both the deck and the living room were unhabitable in the afternoon. We finally had a carpenter build a shed roof extension over the deck, which was/is a fantastic addition to the house. It actually gets a bit cold even on the hottest days out there.

Another option is a motorized awning. We have one of those off the second storey deck outside of our master bedroom. Same issue with sun, and that works very well too.

Still waiting.

Sunshades of any type should just be temporary solutions. Plant trees. You'll love them and so will the ecology.

We had our gabazo for years; it worked well and blocked a lot of sun, even as the day progressed. There are all kinds of styles/permanent, or not. Hard tops have to be cleared of snow and soft tops have to be replaced approximately every 3 years. Both have to be bolted down. Do your research and anytime summer clearance prices will begin. You can also find lots on Marketplace.

We decided on a pergola last year and although we like the look, it’s a pain to open and close tarp, as wind and rain come into play. It also doesn't protect as much from the sun on the outer edges.

IMO the OP has left the building .