Why free standing? Do you want one with a solid roof? I'd do a proper pergola (no roof) and tie it into the existing patio roof. Then use a shade cloth or have vines grow up and over it. For me, the stone on your house are calling for grape vines. You'd also only add 2 posts for a less cluttered look.

The extension looks too shallow depth for much of anything. Draw the whole space and add the feet and inches to that drawing. Clean up the surrounding area, move the grill from literally on top of the door entrance. The step should have run the full width of the extension - actually looks like a hazard and not to any code.

You may be able to squeeze a table with umbrella....but there are other issues , first.

How about a retractable awning? I'm amazed at how much cooler our deck is when we use ours

Consider the height of the pergola. A lower-profile pergola with open sides may obstruct the view less than a taller, more solid structure. Consult with a professional pergola builder.

What direction is this facing?

You could add something that has a retractable roof (perhaps canvas) that you can open when you want more light through your windows and then cover the patio and shade the windows a bit when the sun is most intense.

eta: Your stone is very pretty!

The pergola will affect the sun/light getting into the home. That may be a plus given your climate.

I think the retracable awning is agood idea but I would worry about damaging the stone . A solid roof on a pergola would not work IMO . So the canvas top that can be moved is probablty the best since when not needed you still can have the light in the house.

I would get a nice big umbrella

Why not?