  In any quasi-metric space of homogeneous type, Auscher and Hyt\"onen recently
gave a construction of orthonormal wavelets with H\"older-continuity exponent
$\eta>0$. However, even in a metric space, their exponent is in general quite
small. In this paper, we show that the H\"older-exponent can be taken
arbitrarily close to 1 in a metric space. We do so by revisiting and improving
the underlying construction of random dyadic cubes, which also has other
applications.
