The specific features of scattering of the Gaussian wave packet by the simplest of all reflectionless potentials, the Sech-squared well, are considered. The results of numerical computation show that a wave packet that has passed through this potential overtakes a free wave packet and is both higher and narrower than a free wave packet; in other words, the scattered wave packet accelerates inside the well and preserves its shape better. Comparisons with scattering by the square well and by a series of reflectionless potentials are also presented. Possible applications of the Sech-squared as well as other reflectionless potentials are discussed.