Use a Semi-Neutral Grip
Hand position and the use of strategic grip can set the stage for a more effective and pain-free static and dynamic shoulder position, especially in pressing variations.
A neutral grip (palms face one another) places the shoulders in a safer position for both upper body pressing and pulling. What we sometimes forget is that hand positions aren't as black and white as a traditional pronated grip vs. a neutral grip. Sometimes, the best position is somewhere in the middle.
Semi-Neutral Dumbbell Bench Press
Due to every person having unique anthropometry, there's never going to be a cookie-cutter hand position for pressing that will take all the pain away for 100% of the population. But the majority of lifters fall somewhere between a neutral grip and a semi-pronated grip. Finding the exact hand position that feels best for you is important, and a few degrees of pronation or supination of the hands can make all the difference.
Once you find your perfect hand position, realize that this position doesn't have to always be the same. The natural movement, torque output, and stabilization that come from adjacent joints working together is a pivotal aspect of pressing pain-free. As long as you start out in a semi-neutral grip and stay somewhere within that angle for each end range of a press, you'll be well on your way to pressing pain-free.
About 45 Degrees
To start off with this hand position, think of a 45-degree angle of the dumbbell handles. This is halfway between the neutral grip and traditional grip. You may deviate slightly from here, but see how it feels before you go off the deep end in any direction.