Well, Ive done a strict military with 375 if that lends any sort of credibility to my advice.
The best thing I ever did for taking up my overhead pressing strength was a speed day, done pretty much like the way you described. On top of that I started adding resistancebands which made things even better and they helped me get through my sticking point , which was three-five inches off the shoulders.
I also would suggest overhead lockouts for assistance work and at the end of the workout do isometric pushes (I do them in the powerrack with three different pin setting at the end of an upper body workout) , the way CT describes at three different positions close to your sticking point, the radiation effect should take care of the rest.
And of course always start your workout with the movement you want to improve.