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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 acombination of HOMFLYPT polynomial of knots obtained by certain band sum of thelink components, if all $\bar{\mu}$-invariants of length $\leq k$ vanish. Theyalso 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 $2k+2$ byadding correction terms. The correction terms can be given by a combination ofHOMFLYPT polynomial of knots determined by $\bar{\mu}$-invariants of length$k+1$. In particular, for any 4-component link the $\bar{\mu}$-invariants oflength 4 are given by our formula, since all $\bar{\mu}$-invariants of length 1vanish.

Milnor invariants of length $2k+2$ for links with vanishing Milnor invariants of length $\leq k$