We report the fabrication from a hard polymer of lamellar structures that act as base support planes for high-aspect ratio nanofiber arrays. We experimentally show that nanofiber arrays on lamellae can adhere to both planar and nonplanar surfaces, exhibiting 5 times greater shear strength on a 100 mum peak-to-peak grating than the arrays without the lamellar support structure. The observed behavior on nonplanar surfaces is attributed to the high compliance of the lamellar flaps. The compliance of the combined lamellae and nanofiber arrays is measured to be about 160 times higher than nanofiber arrays on a flat nonlamellar backing layer.