  Connecting double quantum dots via a semiconductor-superconductor hybrid
segment offers a platform for creating a two-site Kitaev chain that hosts a
pair of "poor man's Majoranas" at a finely tuned sweet spot. However, the
effective couplings, which are mediated by Andreev bound states in the hybrid,
are generally weak in the tunneling regime. As a consequence, the excitation
gap is limited in size, presenting a formidable challenge for using this
platform to demonstrate non-Abelian statistics of Majoranas and realizing
error-resilient topological quantum computing. In this work, we systematically
study the effects of increasing the coupling between the dot and the hybrid
segment. In particular, the proximity effect transforms the dot orbitals into
Yu-Shiba-Rusinov states, forming a new spinless fermion basis for a Kitaev
chain, and we derive a theory for their effective coupling. As the coupling
strength between the dots and the hybrid segment increases, we find a
significant enhancement of the excitation gap and reduced sensitivity to local
perturbations. Although the hybridization of the Majorana wave function with
the central Andreev bound states increases strongly with increasing coupling,
the overlap of Majorana modes on the outer dots remains small, which is a
prerequisite for potential qubit experiments. We discuss how the
strong-coupling regime shows in experimentally accessible quantities, such as
the local and non-local conductance, and provide a protocol for tuning a
double-dot system into a sweet spot with a large excitation gap.
