The topological property of a gapped odd-parity superconductor is jointly determined by its pairing nodes and Fermi surfaces in the normal state. We reveal that the contractibility of Fermi surfaces without crossing any time-reversal invariant momentum and the presence of nontrivial Berry phase on Fermi surfaces are two key conditions for the realization of higher-order topological odd-parity superconductors. When the normal state is a normal metal, we reveal the necessity of removable Dirac pairing nodes and provide a general and simple principle to realize higher-order topological odd-parity superconductors. Our findings can be applied to design new platforms of higher-order topological superconductors, as well as higher-order topological insulators owing to their direct analogy in Hamiltonian description.