An expansion for the incomplete Gauss sum Sm(x;p)=∑j=0m-1exp(πixjp), p>1 is obtained for x→0+ for values of m corresponding to the principal spiral 1⩽m<M0, M0=(2/px)1/(p-1) (when the terms of the sum are considered as unit vectors in the complex plane). This expansion results from resumming the terms in the expansion obtained in Paris [An asymptotic approximation for incomplete Gauss sums, J. Comput. Appl. Math. 180 (2005) 461–477]. The new expansion is specialised to the quadratic incomplete Gauss sum with p=2 and x=2/N, where N is a large positive integer, and compared with that obtained by Evans et al. [Incomplete higher-order Gauss sums, J. Math. Anal. Appl. 281 (2003) 454–476].