When it comes to building strength, there are 2 huge, controllable factors that determine strength. They are, in order of importance, neuromuscular coordination and muscle size. Neuromuscular coordination is the ability of your brain, nerves, and muscles to work together efficiently to produce a movement pattern. In everyday language we might refer to this as one’s technique or skill. The second key part is muscle size, otherwise known as the cross sectional area of the muscle. Think of muscles as the engine in a car; muscles are what makes the body go. Neuromuscular coordination is like the driver of the car.
While the overhead press can benefit from a rigid core and the legs being screwed into the ground, the majority of the movement is produced by the muscles in the upper body (assuming one does not turn it into a push press). Because there is some confusion as to what role certain muscles have in the overhead press, I have listed all of the major muscles in the upper body and listed their relative contribution to the overhead press. The assumed form is one is standing vertical, a significant layback can dramatically affect the muscles involved. A score of 5 means the muscle is the agonist or the prime mover. A 4 means the muscle is a strong synergist, it will produce actual movement and it will respond to the exercise. A 3 is a weak synergist or a strong stabilizer. The muscle will contribute a little bit to the exercise and it might show some development from the exercise. A 2 means the muscle is a stabilizer during the exercise. It is contracting during the exercise but it is not producing active movement and it will usually not respond to the exercise. However if this muscle is injured it can significantly limit the weight lifted. A score of 1 means the muscle is relatively inactive during the exercise
Note: It is possible that individual variation, biomechanics, and form might make a muscle work either a little bit more or a little bit less involved based on how the lift is perform.
If you believe that increasing the size of your muscles will help improve the overhead press, then focus on training the muscles that received a 3 or more on the above scale. Give it 3, 6, even 12 months of hard training. Track your progress and try to establish your own personal correlation between your muscle size and your performance on the platform.
It is worth noting that while bench press and overhead press performance will be related, it may not be a perfect relationship. There are some great benchers with a relatively poor overhead press and vice versa. Find out your specific relationship between the two lifts (take the overhead press as a percentage of your bench) and then chart what happens with both lifts. Do they both increase at the same rate or not? Does training one seem to have a direct impact on the other? Is training both of them intensely too much for your shoulder girdle? This will help you determine how important the overhead press is to improving your bench press, if that is a primary focus for you.