In this article we study minimal 1-blocking sets in finite projective spaces PG(n,q), n ≥ 3. We prove that in PG(n,q), q = p, p prime, p > 3, h ≥ 1, the second smallest minimal 1-blocking sets are the second smallest minimal blocking sets, w.r.t. lines, in a plane of PG(n,q). We also study minimal 1-blocking sets in PG(n,q), n ≥ 3, q = p, p prime, p > 3, q ≠ 5, and prove that the minimal 1-blocking sets of cardinality at most q + q+ q + 1 are either a minimal blocking set in a plane or a subgeometry PG(3, q).