We prove a prescribed-leftover-chord theorem for Hamiltonian Berge cycles of odd order. Let $C$ be a Hamiltonian Berge cycle on $n=2r+1$ vertices, and let $\mathcal G$ be a set of hyperedges, all of size at least $r$, containing the hyperedges of $C$. If $D\subseteq\{2,\ldots,r\}$ and $|\mathcal G|\ge n+|D|$, then the hyperedges can be reassigned to the adjacent pairs of the same cyclic order so that, for each $d\in D$, a distinct unused hyperedge realizes cyclic distance $d$. Consequently, the odd-order case of the one-extra-edge question of Bailey, Hollars, Li and Luo has an affirmative answer for all $n=2r+1\ge7$, in the convention including Berge cycles of length $2$. The proof combines an additive lemma in $\mathbb Z_{2r+1}$ with an alternating matching exchange.