Have you noticed the little tab on one side of the box? You can push it in. That tab is there to keep the roll of aluminum foil steady. It makes it much simpler to pull out a piece of foil and tear it off neatly.
Simply flip the lid upside down before placing it on your Slurpee cup, and this will create more room for you to enjoy your favorite drink. However, make sure you drink slowly and avoid drinking too quickly, or you might end up getting a brain freeze.