Abstract We report microscopic studies by muon spin spectroscopy of the superconducting properties as a function of chemical and hydrostatic pressure in the cubic ternary intermetallic (Ca _x Sr_1−x )₃Rh₄Sn₁₃ compounds. We find evidence of a quantum critical point at a critical pressure p _c in the superconducting phase, where the superfluid density increases by a factor of two and the superconducting pairing strength displays a pronounced maximum. The enhancement of superconductivity is related to the structural phase transition at p _c , which is accompanied by profound changes of the Fermi surface associated to the suppression of a charge density wave (CDW). The quantum critical point separates a superconducting phase coexisting with CDW from a pure superconducting phase, while in both phases superconductivity has a strong-coupling phonon-mediated BCS-like s-wave character. Together with the related isoelectronic compound (Ca _x Sr_1−x )₃Ir₄Sn₁₃, this system shows that conventional BCS superconductors in the presence of competing orders may display behaviour and characteristics of unconventional superconductors.