For what matroids $M$ can its circuits be indexed $C_1,\ldots C_n$ s.t. $\forall k<n(C_k\cap C_{k+1}\neq\emptyset$)?

Or equivalently for what matroids $M$ is the intersection graph of $\mathscr{C}_M$ connected?

(this is the graph $\small G$ with $\small V(G)=\mathscr{C}_M$ and $\small \{C_1,C_2\}\in E(\mathscr{C}_M)\iff C_1\cap C_2\neq\emptyset$)

Suppose we call matroids with this property "overlapping" then if we denote the graphic matroid of $G$ by $\mathcal{M}(G)$ note that $G$ is $2$-edge connected iff $\mathcal{M}(G)$ is overlapping. In addition if two matroids $M_1$ and $M_2$ are overlapping and some circuit in $M_1$ is not disjoint to some circuit in $M_2$ then the direct sum $M_1\oplus M_2$ is also overlapping, thus as a corollary every overlapping matroid must be a connected matroid.