The picture on the left is the inside of the shirt, so you can see the dart going down the side of the original shoulder seam.
I’ve seen quite a few alteration videos where they separate the sleeve from the body fabric, and reshape a new armhole, and resew. This is definitely one way, but it’s not an easy way.
Instead of taking apart the garment and reshaping the armhole, I just add a pleat (dart tuck) or a traditional dart at the center of each shoulder. There’s no taking the garment apart, and if it doesn’t look as good as you’d like, you can always take it out, and you’re back with the original garment. No harm done to the garment. This method is quite safe.
In this version, I take the shoulder dart all the way down to the chest, so the chest width becomes narrower as well. This is for a shirt that is too big all around. You can even take the dart all the way down to the bottom hem to narrow the hem width as well.
Video here!