This is really just a long comment expanding on Bruno's answer. However, I wanted to include some handy references.

The approach that will probably be most applicable for the purposes of this question are derived from Agol and Lackenby's 6 theorem. In Exceptional Dehn surgery on the minimally twisted five-chain link, Martelli, Petronio, and Roukema check the hyperbolicity of infinite families of (n-1)-cusped manifolds coming from surgery on an (n)-cusped manifold is handled quite nicely. Also, Ichihara and Masai's Exceptional surgeries on alternating knots uses similar methods, but applies them to a different set of manifolds.

Let me expand upon Martelli, Petronio, and Roukema's argument because I believe it will be most directly applicable for you. Suppose $M$ is a three cusped manifold and $M(\alpha_i, \beta,-)$ (i.e. filling the first cusp along the slopes $\alpha_i$ and the second cusp along $\beta$, and the third cusp is unfilled) is non-hyperbolic for an infinite number of distinct $\alpha_i$. Then $M(-,\beta,-)$ is non-hyperbolic. Similarly, there are only finitely many $\beta$ to check for non-hyperbolicity if $M$ hyperbolic. The code (with documentation) associated to the paper is available on Bruno's website and fairly easy to implement and adjust (after installing Snappy).

Finally, for there were tricks that one could use other than the 6 Theorem.

One method is involves keeping track of the distance between pairs of non-hyperbolic fillings. Cameron Gordon's surveys "Dehn Surgery on 3-manifolds" (available the book Low Dimensional Topology) and Dehn filling: a survey (the later deals mainly with closed manifolds) are good places to dive in the literature if you are trying to understand this approach.

A second method directly applied to knot complements is Kadokami's Hyperbolicity and identification of Berge knots of types VII and VIII. This method involves showing a set of knot complements known to be atoroidal are not torus knots showing that the Alexander polynomials of the knots in questions do not overlap with torus knots. Then Thurston's work establishes the hyperbolicity of the knot complements being considered.

This is really just a long comment expanding on Bruno's answer. However, I wanted to include some handy references.

The approach that will probably be most applicable for the purposes of this question are derived from Agol and Lackenby's 6 theorem. In Exceptional Dehn surgery on the minimally twisted five-chain link, Martelli, Petronio, and Roukema check the hyperbolicity of infinite families of (n-1)-cusped manifolds coming from surgery on an (n)-cusped manifold is handled quite nicely. Also, Ichihara and Masai's Exceptional surgeries on alternating knots uses similar methods, but applies them to a different set of manifolds.

EDIT: In the paragraph below, it is assumed that the framing of the peripheral torus and so the labelling of the surgery slopes is fixed. In terms of the question being posted, it would make sense to use a geometric basis, i.e. take a pair of the shortest two linearly independent elements. As Ian Agol's answer points out, the blackboard framing of one cusp of a link and the geometric framing of the same cusp can be wildly different.

Let me expand upon Martelli, Petronio, and Roukema's argument because I believe it will be most directly applicable for you. Suppose $M$ is a three cusped manifold and $M(\alpha_i, \beta,-)$ (i.e. filling the first cusp along the slopes $\alpha_i$ and the second cusp along $\beta$, and the third cusp is unfilled) is non-hyperbolic for an infinite number of distinct $\alpha_i$. Then $M(-,\beta,-)$ is non-hyperbolic. Similarly, there are only finitely many $\beta$ to check for non-hyperbolicity if $M$ hyperbolic. The code (with documentation) associated to the paper is available on Bruno's website and fairly easy to implement and adjust (after installing Snappy).

Finally, for there were tricks that one could use other than the 6 Theorem.

One method is involves keeping track of the distance between pairs of non-hyperbolic fillings. Cameron Gordon's surveys "Dehn Surgery on 3-manifolds" (available the book Low Dimensional Topology) and Dehn filling: a survey (the later deals mainly with closed manifolds) are good places to dive in the literature if you are trying to understand this approach.

A second method directly applied to knot complements is Kadokami's Hyperbolicity and identification of Berge knots of types VII and VIII. This method involves showing a set of knot complements known to be atoroidal are not torus knots showing that the Alexander polynomials of the knots in questions do not overlap with torus knots. Then Thurston's work establishes the hyperbolicity of the knot complements being considered.