You will perhaps be interested in the paper "Non-integral toroidal Dehn surgeries" by Gordon and Luecke. Here their Theorem 1.1:

Theorem 1.1. Let $K$ be a hyperbolic knot in $S^3$ that admits a non-integral surgery containing an incompressible torus. Then $K$ is one of the Eudave-Muñoz knots $k(l, m, n, p)$, and the surgery is the corresponding half-integral surgery.

Immediately before the theorem statement they remark that the Eudave-Muñoz knots

... are the only known examples of non-trivial, non-integral, non-hyperbolic surgeries on hyperbolic knots.

You will perhaps be interested in the paper "Non-integral toroidal Dehn surgeries" by Gordon and Luecke. They prove:

Theorem 1.1. Let $K$ be a hyperbolic knot in $S^3$ that admits a non-integral surgery containing an incompressible torus. Then $K$ is one of the Eudave-Muñoz knots $k(l, m, n, p)$, and the surgery is the corresponding half-integral surgery.

Almost immediately before the statement they remark that the Eudave-Muñoz knots

... are the only known examples of non-trivial, non-integral, non-hyperbolic surgeries on hyperbolic knots.