Most contact models addressing the solution of the tangential problem assume an elliptical shape of the contact patch which is derived from the Hertz theory. In particular, the linear theory of rolling contact developed by Kalker is based on this assumption. However, it is well known that in many practical cases the shape of the contact region formed between the rail and a railway wheel can be far from elliptic. Recently, the Simple Double-Elliptic Contact (SDEC) shape has been proposed as an approximation of a wide variety of non-elliptic patches. In this paper, a formulation of the linear theory of rolling contact for a SDEC shape is proposed. The coefficients of the original linear theory for elliptic patches are maintained and four new coefficients are introduced to consider the effect of non-ellipticity. The coefficients of the theory are derived numerically using software CONTACT. Then, an extension of FASTSIM for a SDEC region is introduced and the obtained creep force curves under various conditions are compared to existing formulations of FASTSIM, using results from software CONTACT as a reference. The comparisons show that the proposed model is as fast as FASTSIM and provides more accurate solutions than existing versions of FASTSIM.