Approximation of rectifiable sets

Question

Let us say that I have a $H^d$ measurable, $H^d$ finite set $E \subset \mathbf{R}^n$ such that there exists a sequence of $C^1$ submanifold $(S_i)$ of $\mathbf{R}^n$ such that \begin{equation} H^d(E \setminus \bigcup_i S_i) = 0. \end{equation} Then is it true that for $H^d$-a.e. $x \in E$, there exists an index $i$ such that \begin{equation} \lim_{r \to 0} r^{-d} H^d(B(x,r) \cap (E \Delta S_i))=0 \ ? \end{equation} Here $\Delta$ is the symmetric difference, i.e. \begin{equation} (E \Delta S) = (E \setminus S) \cup (S \setminus E). \end{equation}