The topology of isosurfaces changes at isovalues of critical points, making such points an important feature when building contour trees or Morse-Smale complexes. Hexahedral elements with linear interpolants can contain additional off-vertex critical points in element bodies and on element faces. Moreover, a point on the face of a hexahedron which is critical in the element-local context is not necessarily critical in the global context. In ‘`Exploring Scalar Fields Using Critical Isovalues’' Weber et al. introduce a method to determine whether critical points on faces are also critical in the global context, based on the gradient of the asymptotic decider in each element that shares the face. However, as defined, the method of Weber et al. contains an error, and can lead to incorrect results. In this work we correct the error.