The most important thing to remember for kids is it only works when they are having fun. As long as they are having fun they keep learning how to do puzzly kind of things.
What you can do it buy slightly harder puzzles and see if they are still having just as much fun. At first you can help them getting started by handing them pieces that you can actually place instead of having to find pieces themself. (the big cognitive jump going from 9/16 piece puzzles to higher is that you will have pieces that won't fit to anything else you already have). Another way is to point to specific pieces they need to search for.
doing harder puzzles together with your kid makes keep it fun and as long as they are having fun they keep learning. (If they get used to it you can reduce the amount of direction you give them).
I can recommend the big floor puzzles that contain 25+ pieces.
What you want them to learn is that doing new things (together with you) is fun.
That way when they are older it becomes easy to introduce letters/reading/math.
Be careful with making something hard now. If they get discouraged learning at a later age won't be fun anymore.