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We show that the electron-phonon coupling (EPC) in many materials can be significantly underestimated by the standard density-functional theory (DFT) in the local-density approximation (LDA) due to large nonlocal correlation effects. We present a simple yet efficient methodology to evaluate the realistic EPC, going beyond the LDA by using more advanced and accurate GW and screened-hybrid-functional DFT approaches. The corrections that we propose explain the extraordinarily high superconducting temperatures that are observed in two distinct classes of compounds—the bismuthates and the transition-metal chloronitrides—thus solving a 30-year-old puzzle. Our work calls for the critical reevaluation of the EPC of certain phonon modes in many other materials, such as cuprates and iron-based superconductors. The proposed methodology can be used to design new correlation-enhanced high-temperature superconductors and other functional materials that involve electron-phonon interaction.