I remember to have folded that Jack-in-the-box too, back in the 1990's ...
To me, it's an either-or story. When you fold it so that it stays closed, then Jack won't pop out anymore when the lid is opened. Either it will pop, and the box won't stay closed, or the box will stay closed, but it won't pop anymore.
The box lid has zero friction to the box' walls (or at least only veryvery little) so there is no in-between.
Back then, I had made a paperbelt to wrap around the box.