I am pretty sure that you can buy clips that attach to the screen and to the window frame, to keep the screen from popping out. My screens have this, but it is part of the screen itself. I would think you could find some that could be added to an existing screen. I actually used paper clips on one of the screens where the clips broke off, I untwisted it, wedged one end into the frame of the screen, and bent the other end around the frame of the window.
One of my cats got through the corner of the window by pushing the screen out. It was a second story window, and I did not even know he was outside for a few hours, until I heard him meowing. He was hiding under the minivan completely freaked out (he is indoor only). That's when I noticed the broken clips on the window and put in the paperclips. That was a year ago and the clips are still working.
Try your local hardware store.