  J.-B. Meilhan and the second author showed that any Milnor
$\bar{\mu}$-invariant of length between 3 and $2k+1$ can be represented as a
combination of HOMFLYPT polynomial of knots obtained by certain band sum of the
link components, if all $\bar{\mu}$-invariants of length $\leq k$ vanish. They
also showed that their formula does not hold for length $2k+2$. In this paper,
we improve their formula to give the $\bar{\mu}$-invariants of length $2k+2$ by
adding correction terms. The correction terms can be given by a combination of
HOMFLYPT polynomial of knots determined by $\bar{\mu}$-invariants of length
$k+1$. In particular, for any 4-component link the $\bar{\mu}$-invariants of
length 4 are given by our formula, since all $\bar{\mu}$-invariants of length 1
vanish.
