Question

This is in response to Vectornaut's question about wrapping with a thin rectangle. In Geometric Folding Algorithms: Linkages, Origami, Polyhedra, it is argued (Theorem 15.2.1, p.236) that any polyhedron can be covered with a thin strip with arbitrarily small surface area beyond that of the polyhedron. A polyhedral approximation to a cylinder then yields the claim. This is Figure 52.2 (p.234), which gives some idea of switchback turns of the strip used in the argument:
          
Here is a link to the original 1999 paper by Demaine, Demaine, and Mitchell, "Folding Flat Silhouettes and Wrapping Polyhedral Packages: New Results in Computational Origami": link.