Using the class $\delta$ of all singular $R$-modules, we introduce and characterize $\delta$-lifting modules, $\delta$-supplements and $\delta$-coclosed submodules, and give some equivalent conditions to characterize an amply $\delta$-supplemented module $M = \oplus_{i\in I}M_i$ to be $\delta$-lifting. We introduce relatively $\delta$-small projective to give some sufficient conditions that a finite sum of $\delta$-lifting modules is $\delta$-lifting. We also show that $R$ is $\delta$-perfect ($\delta$-semiperfect) if and only if every (finitely generated) projective right $R$-module is $\delta$-lifting.