  In this work we study the impact of recent findings regarding
non-perturbative corrections in the three-jet region to $e^+e^-$ hadronic
observables, by performing a simultaneous fit of the strong coupling constant
$\alpha_s$ and the non-perturbative parameter $\alpha_0$. We extend the
calculation of these power corrections, already known for thrust and
C-parameter, to other $e^+e^-$ hadronic observables. We find that for some
observables the non-perturbative corrections are reasonably well behaved in the
two-jet limit, while for others they have a more problematic behaviour. If one
limits the fit to the three-jet region and to the well-behaved observables, one
finds in general very good results, with the extracted value of $\alpha_s$
agreeing well with the world average. This is the case in particular for the
thrust and $C$-parameter for which notably small values of $\alpha_s$ have been
reported when non-perturbative corrections have been computed using analytic
methods. Furthermore, the more problematic variables are also well described
provided one stays far enough from the two-jet limit, while in this same region
they cannot be described using the traditional implementation of
power-corrections based on two-jet kinematics.
