Don’t know what you do in the Army, but when I was in SpecWar in the Navy, pushups meant shit other than bragging rights. At 210ish and 6’, I was typically hitting mid-50s. Guys (mostly senior to me) would talk shit that day, and then by the time we had showered and work started it was forgotten. Sure, maxing out your PT score looks good on your evals, but in my experience it really didn’t mean much. Army may be different, I know your advancement is different from ours so it may have a bigger effect, or could be the thing that bumps you up a touch, everything else being equal.
That being said, when I did want to work on my pushups for some reason, I would just train them as an extra workout at some point (usually at night, since I lifted in the mornings). I typically used a website that gives a pushup workout promising to get you to 100 pushups, in a 3x per week setup for 6 weeks. They also have them for pullups and situps (even though I know Army guys don’t do pullups, HA). I would do this for a few weeks before the semi-annual test, and then drop them again until next time I needed to work on them. Which wasn’t often.