Blocking semiovals and the determination of their (minimum) sizes constitute one of the central research topics in finite projective geometry. In this article we introduce the concept of blocking set with the $r_\infty$-property in a finite projective plane $\text{PG}(2,q)$, with $r_\infty$ a line of $\text{PG}(2,q)$ and $q$ a prime power. This notion greatly generalizes that of blocking semioval. We address the question of determining those integers $k$ for which there exists a blocking set of size $k$ with the $r_\infty$-property. To solve this problem, we build new theory which deeply analyzes the interplay between blocking sets in finite projective and affine planes.