Determining shortest vector between a ray and an axis-aligned 3D bounding box (correcting a near-miss)
If you know the rounding error, you could subtract the error from your vectors so that rays would overlap the bounding box. This could be done in your intersection tests.

In 3D, computing shortest distance between a vector and face of a bounding box would be another way to shift the vector to overlap the bounding box. But I doubt this approach would work well as it essentially tries to fix a rounding problem.