A scheme for the systematic improvement of the mapping mechanism of the mapped Fourier method (MFM) is presented, Older schemes, which relied on the WKB approximation to build the adapted grid placed very few points close to the classical turning points of a potential, which deteriorates the accuracy of the sampling of the wave functions that are represented on the grid. A numerical example at the end of the Letter will prove the efficiency of the new scheme. (C) 2002 Elsevier Science B.V. All rights reserved.