When a water drop impinges on a flat superhydrophobic surface, it bounces off the surface after a certain dwelling time, which is determined by the Rayleigh inertial-capillary timescale. Recent works have demonstrated that this dwelling time (i.e., contact time) is modified on curved superhydrophobic surfaces, as the drop asymmetrically spreads over the surface. However, the contact time on the curved surfaces still remains poorly understood, while no successful physical model for the contact time has been proposed. Here, we propose that the asymmetric spreading on the curved surface is driven by either the Coanda effect or inertia depending on the ratio of the drop diameter to the curvature diameter. Then, based on scaling analysis, we develop the contact time model that successfully predicts the contact time measured under a wide range of experiment conditions such as different impact velocities and curvature diameters. We believe that our results illuminate the underlying mechanism for the asymmetric spreading over the curved surface, while the proposed contact time model can be utilized for the design of superhydrophobic surfaces for various thermal applications, where the thermal exchange between the surface and the water drop occurs via a direct physical contact.