We use scanning tunneling spectroscopy (STS) experiments and first-principles density functional theory (DFT) calculations to address a fundamental question of how quantum well (QW) states for electrons in a metal evolve spatially in the lateral direction when there is a surface step that changes the vertical confinement thickness. This study reveals a clear spatially dependent, nearly continuous trend in the energetic shifts of quantum well (QW) states of thin Ag(111) film grown on Cu(111) substrate, showing the strongest change near the step edge. A large energetic shift equaling up to  200 meV with a lateral extension of the QW states of the order of  20 Å is found, even though the step-edge is atomically sharp as evidenced by a line scan. The observed lateral extension and the nearly smooth transition of QW states are understood within the context of step-induced charge oscillation, and Smoluchowski-type charge spreading and smoothing.