Recent experiments announced an observation of topological superconductivity and Majorana quasiparticles in Shiba chains, consisting of an array of magnetic atoms deposited on top of a superconductor. In this work we study helical Shiba chains and generalize the microscopic theory of subgap energy bands to a regime where the decoupled magnetic impurity energy and the hybridization of different impurity states can be significant compared to the superconducting gap of the host material. From exact solutions of the Bogoliubov-de Gennes equation we extract expressions for the topological phase boundaries for arbitrary values of the superconducting coherence length. The subgap spectral problem can be formulated as a nonlinear matrix eigenvalue problem from which we obtain an analytical solution for energy bands in the long coherence length limit. Physical consequences and departures from the previously obtained results in the deep dilute impurity limit are discussed in detail.