Do SS. Your pushups will increase without even trying, if you follow the program.
Why do you need to do 20+ pushups?
Looking to start this program on Monday but I also need to increase my pushups for a sub 5 to 20+ in the next few months. I found the 100 pushup challenge online and was going to use that program before doing SS on M/W/F. Is this too much? Should I ONLY do one or the other? I am 190, skinny as hell.
Do SS. Your pushups will increase without even trying, if you follow the program.
Why do you need to do 20+ pushups?
PT tests for police. 18 is minimum but I obviously want to train for more in case they don't count some for bad form or whatever.
Is it realistic to expect gains from sub 5 to 20+ in a month or 2? I guess it depends on making sure I stick to the plan and not alter it. Will I hurt anything by doing at least a few pushups as warm ups?
I think you need to get stronger. Pushups are pretty easy, strength-wise. Doing a set of 18 will be a conditioning test, at that point, but you can work on that a few weeks out.
It might be the case that 3 sets of 5 on squats gets you conditioned enough.
A few years ago, when I wanted to increase my pushups, I did something similar to this https://docs.google.com/document/d/1.../edit?hl=en_US
It worked well. Once you can get a few it is easy to increase them.
I don't think they'll harm you as a warmup. Good luck!
If you want to be able to do a lot of pushups, do a lot of pushups.
Pretty straightforward.
Karl is right that being stronger overall will help you get there (the bigger your bench, the easier they will be), and with your other PT skills for your test.
But if you want to be able do a lot of pushups, do a lot of pushups.