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An intersecting $r$-uniform straight line system is an intersecting linearsystem 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 $r$-uniformstraight 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 asubset of lines $R\subseteq\mathcal{L}$ such that every triplet of differentelements of $R$ does not have a common point.  In this paper, we improve such upper bound if the intersecting $r$-uniformstraight line system satisfies $r=\nu_2$. This result has immediateconsequences for some questions given by Oliveros et al. [D. Oliveros, C.O'Neill and S. Zerbib, \emph{The geometry and combinatorics of discrete linesegment hypergraphs}, Discrete Math. {\bf 343} (2020), no. 6, 111825].

On transversal numbers of intersecting straight line systems and intersecting segment systems