  This is the second of two papers investigating for which positive integers
$m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$
(the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first
part, the sizes of maximal antichains have been characterized. Here we provide
an alternative construction with the benefit of showing that almost all sizes
of maximal antichains can be obtained using antichains containing only $l$-sets
and $(l+1)$-sets for some $l$.
