An exterior set with respect to the hyperbolic quadric Q +(2 n − 1, q) of PG(2 n − 1, q), n ⩾ 2, is a set X of points such that each line joining two distinct elements of X has no point in common with Q +(2 n − 1, q). Then |X| ⩽ (q n − 1) (q − 1) . If |X| = (q n − 1) (q − 1) , X is called a maximal exterior set (MES). The existence of a MES of Q +(2 n − 1, q) implies the existence of a MES of Q +(2 m − 1, q), 2 ⩽ m ⩽ n. All MES of Q +(3, q) were determined. Further, it was proved that Q +(2 n − 1, q), with n ⩾ 3, q ∉ {11, 23, 59} and ( n, q) ≠ (3, 2), has no MES, and that Q +(5, 2) has a unique (up to a projectivity) MES. Here we prove that Q +(2 n − 1, q), n ≠ 3 and q odd, has no MES, thus giving the complete classification of all MES.