  We continue the previous study on the existence of different intermediate
Hamiltonians in type A N-fold supersymmetric systems and carry out an
exhaustive investigation on the N=3 case. In contrast with the N=2 case, we
find various patterns in the existence of intermediate Hamiltonians due to the
presence of two different intermediate positions in a factorized type A 3-fold
supercharge. In addition, all the N=3 models are strictly restricted to at most
elliptic type, which enables us to make the complete classification of the
systems which admit intermediate Hamiltonians. Finally, we show realizations of
third-order parasupersymmetry and variant generalized 3-fold superalgebras by
such systems.
