  Monodromy groups, i.e. the groups of isometries of the intersection lattice
L_X:=H_2/torsion generated by the monodromy action of all deformation families
of a given surface, have been computed in math.AG/0006231 for any minimal
elliptic surface with p_g>q=0. New and refined methods are now employed to
address the cases of minimal elliptic surfaces with p_g+1>q>0. To this end we
find explicit families such that any isometry is in the group generated by
their monodromies or does not respect the invariance of the canonical class or
the spinor norm. The monodromy action is moreover shown to act by the full
symplectic group on the first homology modulo torsion.
