I hate to break the bad news to you, but if you suddenly increase your pushups all at once.. like doubling them, then you might start to get really sore, so you may or may not be able to continue to increase, or if you do too many sets in a row, you might get more sore from that, so even though you might improve your pushups, you also might have some days that you are even more sore than you had been on previous days.
Maybe this will all work out for you, but sometimes there can be advantages in spreading out your pushups throughout the whole day, especially if you are going to continue to want to do several sets in each day and try to continue to do them every single day (even if some days might be higher numbers of pushups and other days might be lower number of pushups, but as you build, you still should be able to continue to increase your overall quantity of pushups, as long as you do not end up injuring yourself or otherwise overdoing it). It is like investing into bitcoin. You want to be aggressive, but not to overdo it otherwise you might end up injuring yourself
You're right after hitting those numbers of pushups all at once, I began to feel some sore in my shoulders which endup slowing down my 24-04-15 pushups, I endup hitting 50 pushups only , I will tried pushing further but I couldn't due to the disturbance of the sore In my shoulders. So I've decided to keep to your advice in spreading out my pushups. Because Doing so will be able to hit some nice numbers of pushups (due to fact that I've gotten some nice stamina in my pushups lately ) without over doing it or over pushing my self. Like I've already hit 20 pushups in my first set of today and later in the day I'm going hit another. Let's me use this medium to drop my report :
100k,I_anime,59,3792,24-04-15

You know we also need to be cautious of the signs our body gives us when we are overly stressed, those sores must mean something and that means you need to reduce your push up amount for that time, spreading out your pushups in such cases would be a good option and maybe taking some pain killers to ease yourself if there are that bad, besides it's all about been fit and healthy at once.
For some days now I've not been well and it even affected my activity on the forum as well as other areas, it was just a simple malaria and typhoid common in my area, but that didn't stop from playing football cause it helps me recover fast and I think sweating too, although I was still able to do up to 20 pushups during that time and I'll give account for all my absent time.
100k,Troytech,20,1390,14/04/2024
100k,Troytech,30,1420,15/04,2024
For the both days I was absent

As soon as I recovered I was able to bounce back to 100 pushups although I had to split into several sets this morning, about 20 each set, for 5 rounds.
100k,Troytech,100,1520,16/04,2024.