Your example for $R=\mathbb{Z}$ (along with the proof) generalizes verbatim to the case $R$ a principal ideal domain and $m=(p)$ with a prime element $p$.

As another example let $R$ be a commutative Noetherian local domain that is not complete. Then also $Hom_R(\hat{R},R)=0$. A concrete example is $R=k[x_1,...,x_n]_{(x_1,...,x_n)}$ with $k$ a field.

This was shown by [Aldrich et. al: Derived Functors of Hom Relative to Flat Covers. Math. Nachr. 242(2002), 17-26], Lemma 3.3.

Added: The following is used in the proof of the cited result:

**Lemma:** If $R$ is a commutative Noetherian local ring then $\hat{R} \to Hom_R(\hat{R},\hat{R}),\;r \mapsto r \cdot id$ is an isomorphism of $R$-modules.

Proof: Let $E=E(R/m)$ be the injective hull of the residue field. By [Brodman, Sharp: Local Cohomology. Theorem 10.2.11] the map $\hat{R} \to Hom_R(E,E), \; r \mapsto r\cdot id_E$ is an isomorphism of $R$-modules (this is sometimes considered as part of Matlis duality). Hence we have $R$-module isomorphisms
$$\begin{array}{lll}
Hom_R(\hat{R},\hat{R}) & \cong & Hom_R(\hat{R},Hom_R(E,E))\cong Hom_R(\hat{R} \otimes_R E,E) \newline
& \cong & Hom_R(E,E) \cong \hat{R}
\end{array}$$
where the 3rd isomorphism uses $\hat{R} \otimes_R E \cong E$ (see [Matsumura: Comm. ring theory. Proof of 18.6 (iii)]. Composing the isomorphisms on element level now yields the map in the lemma.