Certain double covers of cubic surfaces

Let $S$ be a smooth cubic surface in $\mathbb{P}^3$. I would like to understand that variety $V$ that parametrizes lines $\ell$ such that $\ell \cdot S=3P$ with $P \in S$. At any point $P \in S$, let $\Pi_P$ be the tangent plane, and let $\Gamma_P=\Pi_P \cap S$. Generically, $\Gamma_P$ is a plane cubic with a node at $P$ and therefore two tangents at $P$. Each of these satisfies $\ell \cdot S=3P$. This leads us to the fact that $V \dashrightarrow S$ is a double cover. I wanted to know if $V$ has a name, and also what can we say about it.

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For any surface of degree at least three we can consider more generally a similar construction of the set of lines $\ell$ with $\ell \cdot S \ge 3$. This is called the asymptotic double cover of $S$. See
Surfaces in $P^3$ over finite fields, in Topics in Algebraic and Noncommutative Geometry: Proceedings in Memory of Ruth Michler, C. Melles et al. eds., Contemporary Math. 324 (2003) 219-226.