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In this paper we present an improved RI-type prescription appropriate for the non-perturbative renormalization of gauge invariant nonlocal operators. In this prescription, the non-perturbative vertex function is improved by subtracting unwanted finite lattice spacing ($a$) effects, calculated in lattice perturbation theory. The method is versatile and can be applied to a wide range of fermion and gluon actions, as well as types of nonlocal operators. The presence of operator mixing can also be accommodated. Compared to the standard RI' prescription, this variant can be recast as a supplementary finite renormalization, whose coefficients bring about corrections of higher order in $a$; consequently, it coincides with standard RI' as $a\to 0$, however it can afford us a smoother and more controlled extrapolation to the continuum limit. In this proof-of-concept calculation we focus on nonlocal fermion bilinear operators containing a straight Wilson line. In the numerical implementation we use Wilson/clover fermions and Iwasaki improved gluons. The finite-$a$ terms were calculated to one-loop level in lattice perturbation theory, and to all orders in $a$, using the same action as the non-perturbative vertex functions. We find that the method leads to significant improvement in the perturbative region indicated by small and intermediate values of the length of the Wilson line. This results in a robust extraction of the renormalization functions in that region. We have also applied the above method to operators with stout-smeared links. We show how to perform the perturbative correction for any number of smearing iterations, and evaluate its effect on the power divergent renormalization coefficients.