For a given nonempty subset L of the line set of PG(3, q), a set X of points of PG(3, q) is called an L-blocking set if each line in L contains at least one point of X. Consider an elliptic quadric Q−(3, q) in PG(3, q). Let E (respectively, T, S) denote the set of all lines of PG(3, q) which meet Q−(3, q) in 0 (respectively, 1, 2) points. In this paper, we characterize the minimum size L-blocking sets in PG(3, q), where L is one of the line sets S, E ∪ S, and T ∪ S.