  Let $P$ be a path graph of $n$ vertices embedded in a metric space. We
consider the problem of adding a new edge to $P$ such that the diameter of the
resulting graph is minimized. Previously (in ICALP 2015) the problem was solved
in $O(n\log^3 n)$ time. In this paper, based on new observations and different
algorithmic techniques, we present an $O(n\log n)$ time algorithm.
