As you're managing to keep the coop a bit cooler than 'outdoors in the shade' temps (at least at the time you looked at the thermometer), I'd say you're already doing good AS LONG AS that's the hottest it gets inside the coop. The thing to beware of is that an underventilated coop can stay cooler in the morning but then get *hotter* in late afternoon into early evening. A max-min thermometer in the indoor part of the coop, inspected daily, would be a real good idea. I'm not saying your coop IS underventilated, it may well be just fine -- all I'm saying is, b/c it is largely enclosed, it is worth keeping an eye on, on an ongoing basis.
Whether 90 in the coop is cool enough for your chickens (which is really the main issue) is a matter for your particular chickens to tell you. Watch their behavior. If they are acting hot -- digging deep holes to sit in, wings out a bit from bodies, mouths open and panting, that kind of thing, then they are having trouble with the heat and it'd be wise to provide a fan or frozen ice jugs or something like that. OTOH if they are just going about their normal business then they are fine at whatever temps you've got.
And really in the final analysis that's what matters -- how your particular chickens are doing. It does vary. So common sense and observation are kind of the bottom line.
P.s. I really like your coop and run, they seem to strike a nice balance between ventilation and shade and protection from elements -- but you might consider adding screws with washers, or screwed-thru battens, to refasten the hardwarecloth, as it would be a shame to learn the hard way that raccoons can pull staplegun staples out.
Good luck, have fun,
Pat