  Let $c$ be a positive constant. Suppose that $r=o(n^{5/12})$ and the members
of $\binom{[n]}{r}$ are chosen sequentially at random to form an intersecting
hypergraph $\mathcal{H}$. We show that whp $\mathcal{H}$ consists of a simple
hypergraph $\mathcal{S}$ of size $\Theta(r/n^{1/3})$, a distinguished vertex
$v$ and all $r$-sets which contain $v$ and meet every edge of $\mathcal{S}$.
This is a continuation of the study of such random intersecting systems started
in [Electron. J. Combin, (2003) R29] where the case $r=O(n^{1/3})$ was
considered. To obtain the stated result we continue to investigate this
question in the range $\omega(n^{1/3})\le r \le o(n^{5/12})$.
