  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
$\Gamma_0^3\approx 0.87$ 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.
