  This is a contribution to the classification problem for dp-minimal
expansions of $(\mathbb{Z},+)$. Let $S$ be a dense cyclic group order on
$(\mathbb{Z},+)$. We use results on "dense pairs" to construct uncountably many
dp-minimal expansions of $(\mathbb{Z},+,S)$. These constructions are
applications of the Mordell-Lang conjecture and are the first examples of
"non-modular" dp-minimal expansions of $(\mathbb{Z},+)$. We canonically
associate an o-minimal expansion $\mathcal{R}$ of $(\mathbb{R},+,\times)$, an
$\mathcal{R}$-definable circle group $\mathbb{H}$, and a character $\mathbb{Z}
\to \mathbb{H}$ to a "non-modular" dp-minimal expansion of $(\mathbb{Z},+,S)$.
We also construct a "non-modular" dp-minimal expansion of
$(\mathbb{Z},+,\mathrm{Val}_p)$ from the character $\mathbb{Z} \to
\mathbb{Z}^\times_p$, $k \mapsto \mathrm{exp}(pk)$.
