Home button won't fit in replacement screen

I'm replacing the (cracked) screen on my iPhone SE, and the home button that I peeled off of my old screen won't fit in the hole of the new screen.

I've tried enough times and enough ways that I'm confident I'm not just sticking it in sideways. The button fits back into the old screen just fine (with no extra effort).

I held the screens up to each other and looked through the cracked screen's hole to see if the new hole is smaller. It looks like it is, but just barely (a fraction of a millimeter, if even that).

Has anyone had this problem before? Is my part defective, or is there something I can do? I'm tempted to try sandpaper, but I'm not sure whether the screen is even made of something that I could even reasonably sand.