When does a matroid $M$ have a set of circuits $\mathcal{C}$ that can be indexed such that $\{C_1,\ldots C_n\}=\mathcal{C}$ and for all $i\leq n$ there exists $j<i$ with $C_i\cap C_j\neq \emptyset$? Or equivalently stated what properties must the matroid $M$ have for the intersection graph of $\mathcal{C}$ to be a connected graph? I.e. when is the graph $G$ with $V(G)=\mathcal{C}$ and $\{A,B\}\in E(G)\iff A\cap B\neq\emptyset$ connected?

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.