Actually, there are a lot of these kinds of knots.
First, let's set some notation. Start with two manifolds $M_1$ and $M_2$ related by a Dehn surgery along an embedded curve $K$ in $M_1$. After removing a neighborhood of K, we glue in a solid torus $T$ along $\partial N(K)$ to obtain $M_2$. We say the core $K'$ of $T$ is the dual knot to $K$ in $M_2$. The dual knots to many Berge knots are (1,1) knots lens space. Ken Baker's work is a great place to find examples of this. Specifically, in this paper
- Kenneth L. Baker, Surgery descriptions and volumes of Berge knots II: Descriptions on the minimally twisted five chain link, Journal of Knot Theory and Its Ramifications 17 No. 09 (2008) pp. 1099–1120, doi:10.1142/S021821650800652X, arXiv:math/0509055
all knots in families III-VI and VIII-XII are hyperbolic knots in $S^3$ that are dual to (1,1) knots in Lens spaces. Forcing the dual knots to be hyperbolic as well.
If you want a more concrete example. The (-2,3,7) pretzel knot is hyperbolic and admits 2 lens space surgeries. (This was first observed by Fintushel and Stern.) The lens spaces are (18,7) and (19,7) and the dual knots to the (-2,3,7) in these lens spaces are (1,1) knots.
