  The escape from a potential well is an archetypal problem in the study of
stochastic dynamical systems, representing real-world situations from chemical
reactions to leaving an established home range in movement ecology.
Concurrently, L{\'e}vy noise is a well-established approach to model systems
characterized by statistical outliers and diverging higher-order moments,
ranging from gene expression control to the movement patterns of animals and
humans. Here, we study the problem of L\'evy noise-driven escape from an almost
rectangular, arctan potential well restricted by two absorbing boundaries. We
unveil analogies of the observed transient dynamics to the general properties
of stationary states of L{\'e}vy processes in single-well potentials. The first
escape dynamics is shown to exhibit exponential tails. We examine the
dependence of the escape on the shape parameters, steepness and height, of the
arctan potential. Finally, we explore in detail the behavior of the probability
densities of the first-escape time and the last-hitting point.
