Sounds like you are doing well by the chickens. Congratulations!
I am guessing that your corral is oriented east-west (narrow ends to east and west). If this is true they may be just fine. They have a 2' strip of shade now - in the winter when the sun is at a low angle - but they should have more shade as the sun gets higher in the sky in the summer. Shading the west side is highly recommended.
Oddly, I have found it helpful to shade the North side as well as the west side, because that hot late afternoon summer sun blasts in from the northwest from April through mid-September.
You could hang a vertical strip of shadecloth on the south side so they have some options about sun vs. partial shade vs. full shade.
On my breeding pens, which are mostly going to get winter use, the sun blasts in the open south side and the pens are only 6' deep so they have no shade for several hours mid-day. I am putting an 18" horizontal strip of plywood across the front starting about 18" off the ground, at the height of a roosting chicken. (The pen roosts are 18" off the ground.) This throws a strip of shade into the breeding pen and also helps protect roosting birds from direct winds coming from the south.
Growing stuff up the wire is great, as long as you can keep the plants alive and leafy. Backup shade sources are always helpful. Plants have a way of defoliating just as their shade is most needed. Mine are always getting stripped by ants. And if the ants don't get them the caterpillars do. I am a big fan of shadecloth