A fifth-order hybrid shock-capturing finite difference scheme (tc-Hybrid) is employed for solving one-dimensional hyperbolic conservation laws on manifolds (SPDEs). Our previously proposed time-continuous embedding approach is employed to transform SPDEs into embedded SPDEs (EPDEs) in an embedding Euclidean space [JSC 93, 84 (2022)]. The spatial gradients are discretized using the fifth-order characteristic-wise weighted essentially non-oscillatory (WENO-Z) and component-wise upwind central finite difference operators, and the ghost cell values are reconstructed using a sixth-order ENO and Lagrange interpolations in the Cartesian computational tube. The tc-Hybrid scheme identifies smooth and non-smooth regions using the robust and accurate trouble-cell detector (RBF shock-detector and Tukey’s boxplot method). This hybridization allows efficient and accurate resolution of fine-scale structures in smooth regions while capturing singular structures (shock, contact discontinuity, and rarefaction waves) in discontinuous regions in an essentially non-oscillatory manner (ENO property). The tc-Hybrid scheme has been tested on a scalar SPDE (Burgers’ equation and Buckley-Leverett problem) and a system of SPDEs (Euler equations with the Sod, Lax, and shock-density wave interaction problems) on one-dimensional manifolds with curved geometries. The results show that the tc-Hybrid scheme achieves fifth-order accuracy for smooth problems, captures singular structures in an ENO manner, and significantly reduces CPU times.