Something to consider: There is a school of thought that says "Aim small, miss small." Right now, you are aiming at the center of a large target (I assume you can't see, and therefore aren't using, a single specific aiming point on the target when you shoot these groups). Even when you add the black/green square overlay target, it's still rather large, overall.
If I might make a suggestion: try using a smaller aiming point to focus and concentrate your group size. Something like a 3" colored circle/sticker, or even a small square of brightly colored masking tape, centered on the target. Line up the sights in the conventional manner, then try to hit the small colored area, not just the large white "kill zone" on the silhouette. Remember to focus on the sights, and let the target (and the specifically colored area on it) blur slightly. You can still find the center of a blurry colored blob, but if the sights are fuzzy/blurry, then your alignment will suffer (and the group size, along with it). Hold as steady as you can while squeezing the trigger gently and smoothly all the way through, and let's see what happens to the group size. Quite possible you'll see some improvement (but probably not a dramatic one, as you're already doing pretty well).