When folding a 3D object from a 2D material like paper, typically only an approximation of the original surface geometry is needed. Such an approximation can effectively be created by a (progressive) mesh simplification approach, e.g. using an edge collapse technique. Moreover, when searching for an unfolding of the object, this approximation is assumed to be fixed. In this work, we take a different route and allow the approximation to change while searching for an unfolding. This way, we increase the chances to overcome possible ununfoldability issues. To join the two concepts of mesh approximation and unfolding, our work combines the edge collapsing mesh simplification technique with a Tabu Unfolder, a robust mesh unfolding approach. We empirically show that this strategy performs faster and that it is more reliable than prior state of the art methods.