  This is a second paper in our ongoing calculation of the
next-to-next-to-leading order (NNLO) QCD correction to the total inclusive
top-pair production cross-section at hadron colliders. In this paper we
calculate the reaction $q\bar q \to t\bar t + q\bar q$ which was not considered
in our previous work on $q\bar q \to t\bar t +X$ due to its phenomenologically
negligible size. We also calculate all remaining fermion-pair-initiated
partonic channels $qq', q\bar q'$ and $qq$ that contribute to top-pair
production starting from NNLO. The contributions of these reactions to the
total cross-section for top-pair production at the Tevatron and LHC are small,
at the permil level. The most interesting feature of these reactions is their
characteristic logarithmic rise in the high energy limit. We compute the
constant term in the leading power behavior in this limit, and achieve
precision that is an order of magnitude better than the precision of a recent
theoretical prediction for this constant. All four partonic reactions computed
in this paper are included in our numerical program Top++. The calculation of
the NNLO corrections to the two remaining partonic reactions, $qg\to t\bar t+X$
and $gg\to t\bar t+X$, is ongoing.
