We consider the first exit point distribution from a bounded domain $\Omega$ of the stochastic process $(X_t)_{t\ge 0}$ solution to the overdamped Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ starting from deterministic initial conditions in $\Omega$, under rather general assumptions on $f$ (for instance, $f$ may have several critical points in $\Omega$). This work is a continuation of the previous paper \cite{DLLN-saddle1} where the exit point distribution from $\Omega$ is studied when $X_0$ is initially distributed according to the quasi-stationary distribution of $(X_t)_{t\ge 0}$ in $\Omega$. The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.