In this work, we present an extension of the classical DORT method for time-harmonic inverse medium scattering problems. This tool enables one to estimate the shape of the unknown scatterers even if they are not necessarily point-like scatterers. This method is described and compared with existing sampling methods. Moreover, a mathematical derivation is provided for its validation, either in a free-space environment or in a metallic enclosure setup. Two-dimensional examples, based on measured data sets acquired in the circular microwave scanner developed at Institut Fresnel, are presented, which show that the method is computationally efficient and robust to noise.