A machine for making non-cylindrical paper container, including wings for wrapping a container blank around a non-cylindrical mandrel. The wings each have a supplement wing portion pivotably attached thereto, and as the wings are moved into the mandrel-engaging position, a mandrel-contacting part of the supplement wing portions that abuts against the mandrel causes those portions to swingingly approach the mandrel. In this manner, the wing portions may move unobstructively past the mandrel as the wings and wing portions wrap the blank against the mandrel.