Degree of irrationality of a very general abelian variety

Consider a very general abelian variety A of dimension at least 3 and an integer 0 < d <= dimA. We show that if the map A(k) -> CH0(A) has a d-dimensional fiber then k >= d + (dimA + 1)/2. This extends results of the second-named author which covered the cases d = 1, 2. As a geometric application, we prove that any dominant rational map from a very general abelian g-fold to P-g has degree at least (3g + 1)/2 for g >= 3, thus improving results of Alzati and the last-named author in the case of a very general abelian variety.