  Given a lower content $d$-regular set in $\mathbb{R}^n$, we prove that the
subset of points in $E$ where a certain Dini-type condition on the so-called
Jones $\beta$ numbers holds coincides with the set of tangent points of $E$, up
to a set of $\mathcal{H}^d$-measure zero. The main point of our result is that
$\mathcal{H}^d|_E$ is not assumed to be $\sigma$-finite; because of this, we
use a certain variant of the $\beta$ coefficient, firstly introduced by Azzam
and Schul in [AS1], which is given in terms of integration with respect to the
Hausdorff content.
