  For a positive integer $d\geq 2$, a family $\mathcal F\subseteq
\binom{[n]}{k}$ is said to be d-wise intersecting if $|F_1\cap F_2\cap
\dots\cap F_d|\geq 1$ for all $F_1, F_2, \dots ,F_d\in \mathcal F$. A d-wise
intersecting family $\mathcal F\subseteq \binom{[n]}{k}$ is called maximal if
$\mathcal F\cup\{A\}$ is not d-wise intersecting for any
$A\in\binom{[n]}{k}\setminus\mathcal F$. We provide a refinement of O'Neill and
Verstra\"{e}te's Theorem about the structure of the largest and the second
largest maximal non-trivial d-wise intersecting k-uniform families. We also
determine the structure of the third largest and the fourth largest maximal
non-trivial d-wise intersecting k-uniform families for any $k>d+1\geq 4$, and
the fifth largest and the sixth largest maximal non-trivial 3-wise intersecting
k-uniform families for any $k\geq 5$, in the asymptotic sense. Our proofs are
applications of the $\Delta$-system method.
