  We continue the study of maximal families W of the Hilbert scheme,
H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a
smooth degree-s surface S containing a line. For s > 3, we extend the two
ranges where W is a unique irreducible (resp. generically smooth) component of
H(d,g)_{sc}. In another range, close to the boarder of the nef cone, we
describe for s=4 and 5 components W that are non-reduced, leaving open the
non-reducedness of only 3 (resp. 2) families for s > 5 (resp. s=5), thus making
progress to recent results of Kleppe and Ottem in [28]. For s=3 we slightly
extend previous results on a conjecture of non-reduced components, and in
addition we show its existence in a subrange of the conjectured range.
