For what matroids $M$ can its circuits be indexed $\small C_1,\ldots C_n$ (not necessarily in a distinct manner) such that $\small \forall 0<i\leq n\exists j<i(C_i\cap C_{j}\neq\emptyset$)?

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 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 $M_1\oplus M_2$ is also overlapping, thus as a corollary every overlapping matroid must be a connected matroid.