Recent developments in the search for topological superconductivity have brought lattices of magnetic adatoms on a superconductor into intense focus. In this work we will study ferromagnetic chains of adatoms on superconducting surfaces with Rashba spin-orbit coupling. Generalizing the deep-impurity approach employed extensively in previous works to arbitrary subgap energies, we formulate the theory of the subgap spectrum as a nonlinear matrix eigenvalue problem. We obtain an essentially analytical description of the subgap spectrum, allowing an efficient study of the topological properties. Employing a flat-band Hamiltonian sharing the topological properties of the chain, we evaluate the Z-valued winding number and discover five distinct topological phases. Our results also confirm that the topological band formation does not require the decoupled Shiba energies to be fine-tuned to the gap center. We also study the properties of Majorana bound states in the system.