  This is a continuation of "Rational families of vector bundles on curves, I".
Let C be a smooth projective curve of genus at least 2 and let M be the moduli
space of rank 2, stable vector bundles on C, with fixed determinant of degree
1. For any k>0, we find all the irreducible components of the space of rational
curves of degree k on M and their maximal rationally connected quotients.
