Origami has many advantages, such as deployability, low mass, and ease of miniaturization; it has potential contributions in many applications. Specifically, origami that has been designed to fold into a target shape can be used in deployable tents, bridges, grippers, packaging, and deformable mirrors for laser communication. This work presents a two-step methodology for optimizing origami designs such that the distance between the bottom side of an origami and its target surface is minimized—the origami will thus lay “on top” of its target surface. The method combines the strengths of rigid origami and non-rigid origami: it starts with a non-foldable rigid origami Miura-ori tessellation that matches the target surface and optimizes this design such that it becomes foldable. Then it uses this crease pattern as the initial design for an optimization cycle that includes the nonlinear N5B8 bar-and-hinge non-rigid origami model and converges to the crease pattern that folds into the target shape, demonstrated in this work with a saddle, dome, and arch. The results show that for an 800 mm by 800 mm design, the folded saddle origami has a maximum distance to its target surface of 24 mm, with the other tested shapes having similar outcomes. This performance will be sufficient for some use cases, but more work is required for high-accuracy applications.