Overhead pressing, or shoulder pressing is one of the most commonly compensated lifts I see. By compensated I mean arching of the low back, bad mechanics of the scapula and looking like they're naturally trying to turn the lift more into an incline press than a straight overhead press.
Continuing to overhead press with these types of mechanics is going to lead to eventual breakdown somewhere. This breakdown can occur in the low back from the constant extension (shear forces leading to destructive movements in your spine) or obviously the shoulder joint.
I've seen it tried to get fixed in many different ways. The most popular one I've seen is the elastic band rotator cuff strengthening. Problem is, as with many "shoulder warm-up" exercises is they don't address the main issue which is overall movement quality. During standing overhead movements common cues of firing your glutes and bracing your core may clean up some of the movement, but it's not only difficult to maintain but poor mechanics will eventually find their way to another area of the body.
Whenever I look to research things of the shoulder, I look to the "shoulder guy" in my industry, Eric Cressy. In his article, Why You Struggle to Train Overhead - And What to Do About It. Instead of addressing your shoulder issue with an elastic band he gives 6 video exercises you can incorporate into your daily warm-up to begin building, not only your quality range of motion, but overhead stability as well.