The paper deals with state estimation of the nonlinear stochastic systems by means of the unscented Kalman filter with a focus on specification of the sigma-points. Their position is influenced by two design parameters the scaling parameter determining the spread of the sigma-points and a covariance matrix decomposition determining rotation of the sigma-points. In this paper, a choice of the scaling parameter is analyzed. It is shown that considering other values than the standard choice may lead to increased quality of the estimate, especially if the scaling parameter is adapted. Several different criteria for the adaptation are proposed and techniques to reduce computational costs of the adaptation are developed. The proposed algorithm of the unscented Kalman filter with advanced adaptation of the scaling parameter is illustrated in a numerical example. (C) 2014 Elsevier Ltd. All rights reserved.