Long-range-corrected range-separated hybrid approximations of the exchange-correlation (XC) energy are proposed through the satisfaction of the local-density linear response, which requires the second-order term of the XC gradient expansion to vanish. The underlying proposition is tested with the asymptotic Perdew-Burke-Ernzerhof (APBE) [L. A. Constantin et al., Phys. Rev. Lett. 106, 186406 (2011)10.1103/PhysRevLett.106.186406], PBE constructed for interfaces (PBEint) [E. Fabiano et al., Phys. Rev. B 82, 113104 (2010)10.1103/PhysRevB.82.113104], and semiclassical generalized-gradient approximation at fourth order (SG4) [L. A. Constantin et al., Phys. Rev. B 93, 045126 (2016)10.1103/PhysRevB.93.045126] functionals by constructing their corresponding exchange holes. Systematic investigation of these long-range-corrected XC hybrids shows improved performance on recovering the local-density linear response. The long-range-corrected XC hybrid based on the APBE functional and satisfying local-density linear response gives promising results compared to the popular LC- for most of the thermodynamical test sets. Most interestingly, the underlying construction and improved performance by these XC hybrids indicate that the proposed methodology can serve as a basis for developing range-separated XC hybrid functionals of greater potentialities in the future, not only long-range-corrected ones but also screened hybrid functionals.