In this work, we derive a new sharp asymptotic equivalent in the small temperature regime $h\to 0$ for the mean exit time from a bounded domain for the non-reversible process $dX\_t=b(X\_t)dt + \sqrt h \, dB\_t$ under a generic orthogonal decomposition of $b$ and when the boundary of $Ω$ is assumed to be \textit{non characteristic}. The main contribution of this work lies in the fact that we do not assume that the process $(X\_t,t\ge 0)$ is \textit{Gibbsian}. In this case, a new correction term characterizing the \textit{non-Gibbsianness} of the process appears in the equivalent of the mean exit time. The proof is mainly based on tools from spectral and semi-classical analysis.