Ordered sequential algorithms for the global minimization of univariate functions over an interval proceed by evaluating this function at successive points corresponding to increasing values of the variable. Extending a recent idea of Wingo, we propose a new ordered sequential algorithm in which step-sizes are such that the function can be approximated locally by a polynomial with a precision of ε. This algorithm explores efficiently the vicinity of the optimum, while other sequential algorithms do not. Computational experiments with several variants of the proposed method and with the algorithm of Vasil'ev-Ganshin are reported on.