This is a common problem; what's happened is you've detached the screen from its mounting frame instead of detaching the screen frame from the phone. Look around the outside edge of your new screen; you see that black frame surrounding it? That's the frame. If you check your old screen you'll see the black frame is missing; now look on the phone itself and you'll see the missing frame.
So the key here is to disconnect the screen, then go through the screen removal process again with your opening pick, only this time get the pick between the frame and the phone. Once you've gotten all the way around, the frame will come off and you can discard it.
Be sure to clean all the old adhesive off and apply a new sheet of precut adhesive and you should be all set to go.