Using potential models we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions we have observed that, at first order, it is necessary to supplement the theory with one finite-range parameter, $Gamma_n^3$, for each specific $n$-level [Kievsky and Gattobigio, Phys. Rev. A {bf 87}, 052719 (2013)]. The value of $Gamma_n^3$ depends on the way the potentials is changed to tune the scattering length toward the unitary limit. In this work we analyze a particular path in which the length $r_B=a-a_B$, measuring the difference between the two-body scattering length $a$ and the energy scattering length $a_B$, results almost constant. Analyzing systems with very different scales, as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of $Gamma_0^3approx 0.87$ for the ground state level. This observation suggests the possibility of constructing a single universal function that incorporate finite-range effects for this class of paths. The result is used to estimate the three-body parameter $kappa_*$ in the case of real atomic systems brought to the unitary limit thought a broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation with the two-body contact $C_2$ at the unitary limit.
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