Biomimetic adhesives with a stiff fibre-reinforced base layer generate strong attachment, even without bioinspired micropatterning of the contact surface. However, current fibre-reinforced adhesive designs are still less versatile with respect to substrate variability than their biological counterparts. In this study, we enhance the comformability of a fibre-reinforced adhesive on curved substrates by adding bioinspired soft backings. We designed and fabricated soft backing variations (polyurethane foams and silicone hydroskeletons) with varying compressive stiffnesses that mimic the soft viscoelastic structures in the adhesive appendages of tree frogs, geckos and other animals. The backings were mounted on a smooth silicone layer enforced with a polyester mesh, and we experimentally investigated the contact area and friction performance of these adhesives on a curved substrate. The results show that the contact area and friction created by a fibre-reinforced adhesive with a soft backing in contact with a non-flat substrate scale inversely with backing stiffness. The integration of stiff fibre-reinforcement with a compressible backing represents an important step in bringing bioinspired adhesives out of the laboratory and into the real world, for example in soft robotic grippers. Moreover, our findings stimulate further research into the role of soft tissues in biological adhesive systems.