The crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. The cone CG, obtained from G by adding a new vertex adjacent to all the vertices in G. Let ϕ s (k) = min { c r (C G) - c r (G) } , where the minimum is taken over all the simple graphs G with crossing number k. Alfaro et al. (SIAM J Discrete Math, 32: 2080–2093, 2018) determined ϕ s (k) ≤ k for k ≥ 3 , and proved ϕ s (3) = 3 , ϕ s (4) = 4 and ϕ s (5) = 5 . In this work, we improve the upper bound for ϕ s (k) and our main result includes the following, slightly surprising, fact: ϕ s (6) = 5 and ϕ s (7) = 6 . [ABSTRACT FROM AUTHOR]