For a non-singular projective toric variety $X$, the virtual logarithmic Tevelev degrees are defined as the virtual degree of the morphism from the moduli stack of logarithmic stable maps $\overline{\mathcal{M}}_{\mathsf{\Gamma}}(X)$ to the product $\overline{\mathcal{M}}_{g,n} \times X^n$. In this paper, after proving the genus $0$ correspondence theorem in this setting, we use tropical methods to provide closed formulas for the case in which $X$ is a Hirzebruch surface. In order to do so, we explicitly list all the tropical curves contributing to the count.