Hyde..yes I would be excluding all other pressing..it would be 100 percent dedication to overhead pressing and id like to stay about the same bodyweight but ive noticed that the overhead press is one lift where I have to gain weight to see any improvement, and no this isn't for a competition I just want to get as strong as I can with this lift just for general strength training.
So we know bulking up (gaining weight for leverage’s sake) isn’t an option. So frequency of pressing is indeed the best bet. It’s going to stimulate muscle protein synthesis where you need directly most often, and more relevant get you more technical work to refine your motor pattern.
That being said, I wouldn’t cut out things that could be direct builders for your weak spot. If you’re going to press 4 days a week for example, neglecting a tool like a push press or 2 board bench if you miss midway up, or 3 board, slingshot, or chain bench if you miss at the very lockout, would be ideal. And the incline pressing can really help your starting strength & your ability to lay back mid lift to grind it out, provided your back is strong enough for that.
If you want to improve strict pressing, doing that often is number one - but don’t be afraid to address a specific weakness as well! I’d rather strict press 3x a week and closegrip bench once than strict press 4x a week, because I know if that goes up so does my strict power - every time.