Let $T\geq1000$ and $X = \exp(\log\log T/\log\log\log T)$. Consider any set $O$ of disjoint open intervals $I$ of length $1/X$, contained in the interval $T\leq t\leq T+e^X$. We prove in this paper, that $\vert\log\zeta(1+it)\vert\leq\varepsilon\log\log T$ in $O$ with the exception of $K$ intervals $I$, where $0<\varepsilon\leq1$ and $K$ depends only on $\varepsilon$.