Maybe a single pass of root-of-3 subdivision would already help? It would insert another "control point" at the middle of the existing triangles. Then you could linearly interpolate along the new refined triangles. That wouldn't really remove the saddle points (if my intuition is correct it would just divide them into smaller ones along the new triangles), but maybe they'll end up more smoothed out and therefore less jarring.

[0] https://www.graphics.rwth-aachen.de/media/papers/sqrt31.pdf