  In \cite{NaTa} Naor and Tao extended to the metric setting the $O(d \log d)$
bounds given by Stein and Str\"omberg for Lebesgue measure in $\mathbb{R}^d$,
deriving these bounds first from a localization result, and second, from a
random Vitali lemma. Here we show that the Stein-Str\"omberg original argument
can also be adapted to the metric setting, giving a third proof. We also weaken
the hypotheses, and additionally, we sharpen the estimates for Lebesgue
measure.
