Noise driven escape from the potential well is the basic component of various noise induced effects. The efficiency of the escape process or time scales matching is responsible for occurrence of the stochastic resonance and (stochastic) resonant activation. Here, we are extending the discussion on how the structure of the potential can be used to optimize the mean first passage time. It is demonstrated that corrugation of the potential can be beneficial under action of the weak Gaussian white noise. Furthermore, we show that the noise tuning can be more effective than shaping the potential. Therefore, action of the tuned additive $\alpha$-stable noise can accelerate the escape kinetics more than corrugation of the potential. Finally, we demonstrate that mean first passage time from a potential well can be a non-monotonous function of the stability index $\alpha$.