Shoulder injuries that lead to pain with overhead movements need to be corrected and then correct overhead biomechanics need to be engrained in order to remain pain free and build shoulder strength.
Hanging is a great way to achieve healthy shoulders, I've noticed a massive improvement in my own shoulder/neck region where I have a history of injury. Over the past few months I have incorporated a mixture of passive; active and dynamic hanging into my training along with spending a lot more time on the rings working pullups; front & back lever regressions and lat activations. Not only has my once often rigid neck/shoulder region softened up but my grip strength has become symmetrical and increased in strength. My grip strength used to be about 36kg on my Right and 32kg on my Left (side of old injury and non-dominant side) and now is an equal 40kg each side.
You don't always have to do endless amounts of traditional external and internal rotator resisted exercises to achieve strong shoulders...in fact, if this is all you are doing you may need to find yourself another practitioner. Your shoulders need to be ready for life and in life we move in many ways and put our shoulders under all sorts of stresses and loads. You need to ensure you have optimum mobility and back this up with a good foundation of reactive stability in order to keep your shoulders healthy in the long term.
My training has also included plenty handstand practice (working my line against the wall); Turkish Get Ups - these too have improved massively, 6 months ago 8kg was the max I felt comfortable with on my Left and now I am an equal 14kg for at least 1-2reps comfortably and the Left feels just as strong as the RIght. Various other cable & kettlebell stabilisation and strength exercises have also formed part of my training.
For a more in depth read about hanging click on this link to Ido Portal's blog on hanging.
Of course, if you have a shoulder injury or have limitations in your shoulders please get assessed first before jumping into anything new.