  The upper estimates for the optimal constants of the multilinear
Bohnenblust--Hille inequality obtained in [J. Funct. Anal. 264 (2013),
429--463] are here improved to: {0.1cm}
  {enumerate} For real scalars: $K_{n}\leq\sqrt{2}(n-1)^{0.526322}$. For
complex scalars: $K_{n}\leq\frac{2}{\sqrt{\pi}}(n-1)^{0.304975}$.{enumerate}
{0.1cm} \noindent We also obtain sharper estimates for higher values of $n$.
For instance, \[ K_{n}<1.30379(n-1) ^{0.526322}\] for real scalars and
$n>2^{8}$ and \[ K_{n}<0.99137(n-1) ^{0.304975}\] for complex scalars and $n >
2^{15}.$
