Being ubiquitously used as anti-adhesive and wound-covering mechanisms, thin films have potential therapeutic uses as cell sheets to target inner organs while navigating narrow environments. A significant challenge to realize versatile films lies in achieving compact storage and efficient transport while ensuring coherency in curvature-bounded environments. In this paper, we propose a folding mechanism of a curved film by using a spiral approach, enabling efficient unfolding and flexible plasters with curved surfaces. Our experiments using gelatin-based films with curved surfaces shows the superior indwelling ability in terms of chromaticity level compared to the conventional planar films, as well as the efficient unfolding in the order of seconds. Our results presents the theoretical and experimental building blocks to realize a versatile class of films which are able to navigate narrow environments, and unfold efficiently and flexibly.