A graph G is super connected, or simply super-κ, if every minimum vertex-cut isolates a vertex. The Kronecker product G₁ × G₂ of graphs G₁ and G₂ is the graph with vertex set V (G₁ × G₂) = V(G₁) × V (G₂) and edge set E(G₁ × G₂) = {(u₁,v₁)(u₂,v₂): u₁u₂ ∈ E(G₁),v₁v₂ ∈ E(G₂)}. In this paper, we prove that P_m × C_n is super-κ for m≥4, n≥3 and n is odd; C_m × C_n is super-κ for m≥5, n≥3 and n is odd.