This paper continues a previous investigation of the three-atom potential (3AP) of xenon, where the number of adjustable parameters has been varied. This time, the parameters are restricted to make the 3AP deviate as little as possible from the multipole expansion at long range, while their number is kept fixed at 26. To find the minimum deviation that is necessary to satisfy the experimental data, a statistical criterion is derived and applied. A few tests show that the 3APs thus obtained possess considerable predictive power. The method yields a 3AP that shows overlap effects at short range and that can be regarded as the ‘best’ many-body potential available for xenon.