I didn't have time to make another frame and cover it in a stiff shell, so I decided to go straight for a stiff shell.
I had some thick corrugated card*, too thick for smooth curves, so I decided to go geometric, and make a truncated cuboid
- six octahedra, in the planes of the faces of a cube, with equilateral triangles filling in the corners.
I used the largest handy sheet of paper (a page of newsprint) to make a square.
Folding into four, then a triangle, and cutting off one point makes a nice octahedron when opened.
I cut two "proper" octahedrons out of the card, then folded the template across the middle to make it narrower - I wanted the piñata to be slightly flatter than a cube.
*Saved from the box that a garden bench came in - Kitewife got fed up with waiting for me to get round to making her one.