  An intersecting $r$-uniform straight line system is an intersecting linear
system whose lines consist of $r$ points on straight line segment of
$\mathbb{R}^2$ and any two lines share a point. Recently, the author [A.
V\'azquez-\'Avila, \emph{On intersecting straight line systems}, J. Discret.
Math. Sci. Cryptogr. Accepted] proved that any intersecting $r$-uniform
straight line system $(P,\mathcal{L})$ has transversal number at most
$\nu_2-1$, with $r\geq\nu_2$, where $\nu_2$ is the maximum cardinality of a
subset of lines $R\subseteq\mathcal{L}$ such that every triplet of different
elements of $R$ does not have a common point.
  In this paper, we improve such upper bound if the intersecting $r$-uniform
straight line system satisfies $r=\nu_2$. This result has immediate
consequences for some questions given by Oliveros et al. [D. Oliveros, C.
O'Neill and S. Zerbib, \emph{The geometry and combinatorics of discrete line
segment hypergraphs}, Discrete Math. {\bf 343} (2020), no. 6, 111825].
