Let d ≥ 2 be an integer, K0 a perfect field such that char(K0)∤d, n > d an integer prime to d, f(x) ∈ K0[x] a degree n monic polynomial without repeated roots, and Cf,d a smooth projective model of the affine curve yd = f(x). Let J(Cf,d) be the Jacobian of the K0-curve Cf,d. We identify Cf,d with its canonical image in J(Cf,d) (such that the infinite point of Cf,d goes to the zero of the group law on J(Cf,d)). We say that an integer m > 1 is (n,d)-reachable over K0 if there exists a polynomial f(x) as above such that Cf,d(K0) contains a torsion point of order m. Earlier we proved that if m is (n,d)-reachable, then either m = d or m ≥ n (in addition, both d and n are (n,d)-reachable). In this paper, we prove the following: If n < m < 2n and if m is (n,d)-reachable over K0, then either d|m or m ≡ n mod d. If either char(K0) = 0, or K0 is infinite and char(K0) > n, then d ⋅ [(n + d)/d] is (n,d)-reachable if and only if n − (d − 1) ⋅ [(n + d)/d] ≥ 0. If char(K0) = 0, then n + d is (n,d)-reachable if and only if d2 − 2d < n. If d = 2 (the hyperelliptic case) and char(K0) = 0, then m is (n,d)-reachable if n + 1 ≤ m ≤ 2n + 1 and m≠2n. (The case when n ≤ m ≤ 3(n − 1)/2 was done earlier by E. V. Flynn.)