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Stalling, or plateauing, is a very common occurrence. It will happen time and time again, and it happens to all of us. Knowing how to deal with it will help you overcome them though, so I'll be focusing on that.
The problem with altering your program
I would caution against altering a tried-and-true program. The numbers and frequencies are chosen with care.
One of the possible problems you will encounter if you increase the volume/frequency of pressing movements only, is that the muscles involved here will overpower your back muscles, resulting in poor posture.
Particularly bad is the idea of a beginner making their own programs. This is a very common mistakes that usually have less-than-ideal results, because programming a good routine requires pretty substantial knowledge in order to not overlook important factors, and avoid exercises that are straight-up bad for you.
Also, beginners who make their own programs tend to place too much focus on the bodypart they have the highest hopes for. For most guys, this would be chest/shoulders and arms, neglecting back and legs.
Greasing the groove
The alteration you suggest isn't really "greasing the groove". GTG is more based on doing exercises throughout the day. Not just by adding a few sets while you're at the gym, but by doing sets every waking hour of every day.
While this could be a solution, it has to be done with a mindset of including every big muscle group. You might be inclined to do pushups under GTG, but then you're again at risk of poor posture. Instead, you should be doing pushups AND pullups AND squats, or some other combination that includes both pressing, pulling, and legs.
Keep it simple. If you have reached a plateau, it's better to find a different program - a different full-body program, that is - in order to challenge your muscles and central nervous system in a new way. As I mentioned, variation is key. And for that reason, programs should be switched out and in, avoiding stagnation.
There is a plethora of different programs out there. Personally, I'm a fan of Wendler's 5/3/1, and I know a lot of people have had great success with Stronglifts 5x5.
Trying one of these for 6-8 months, and then the other, you will probably have an easier time breaking plateaus. And with this, you will gain more experience, find out what works best for your body in particular, and you can have a wiser look next time you hit a plateau. And you will. We all hit them. Knowing how to break them instead of becoming demotivated is very important.