The result of Padrol (Discret Comput Geom 50(4):865–902, 2013) asserts that for every d≥ 4 , there exist 2 ^Ω(nlogn) distinct combinatorial types of ⌊ d/ 2 ⌋ -neighborly simplicial (d- 1) -spheres with n vertices. We present a construction showing that for every d≥ 5 , there are at least 2Ω(n⌊(d-1)/2⌋) such types.