Our entire run is built out with lumber and covered with hog paneling so no aerial predator worries. It has a six-post construction with 2x4 planks overlaying the top that are stapled down with wire.
It's also dug in at the bottom with wire too to deter burrowing predators, but I have dogs in the yard and a six-foot privacy fence so hawks are my biggest predator concern where I'm at (and they are everywhere). Redtail hawks and other small hawks are already attracted to my yard because I feed the local squirrels and I have about six bird feeders.
The whole thing cost six hundred bucks to have someone else build out (just labor, not lumber), so not cheap. But for me it was worth the investment to never have to worry about hawks, and I was too short to take on the work myself. I got a private contractor to do it.
I will say this - absolutely shell out for a top on your run that you don't have to lean to get into, because I used a droopy bird netting to cover my run the first year (plus I only had short four-foot wrought iron fencing), and it was a massive pain in the you-know-what. I'd practically have to crawl in to inspect the coop. It didn't get cleaned as often either because I didn't want to have to slouch-crawl around in the run, especially when it got wet or during winter. I also had a small flighty hen escape through the netting where it had worked loose and get killed by one of the dogs.
Now I can just walk straight into the run and do my thing, no crouching necessary. No escape worries. Major quality-of-life improvement and MUCH easier to maintain. Plus it just looks nice too.