Let the edge of the cube be of length 1 WLOG.

If the point of attachment is a vertex, the center of the top-left circle on the grid illustration , the three faces that meet at that vertex are covered. I've labeled these as Left, Top and Front. By symmetry, any one of the right, bottom or back will exhibit the same coverage, that is, accessible areas. 

Any of the bottom, right or back faces will be the ones to experience any inaccessability; let's take the bottom:

Look at the diagram. The bottom, when folded under from its left position, will have its entire right side edge inaccessible, most deeply toward the back; when folded under from the position at the bottom of the diagram, it's the back edge that will have the zone of inaccessibility along it, most thickly on the right. So the area of inaccessibility is a quadrilateral, curved on two sides, adjacent to the back and right faces. As said, those other two faces will have adjoining quadrilaterals of inaccessibility.

                                           

                                           

When the tether is attached to the center of a face, as in the top-right circle, let's assume it's attached to the top. Clearly the left, right, front and back are accessible. When the ant goes through the left or right faces almost half the bottom is covered. Only two small triangular slivers are left. These two slivers are completely covered by where the ant can go by going through the back or front to the bottom.

         

           

With the tether attached to the center of an edge, say the TF edge in the bottom diagram, clearly the left, top, right and front faces are completely accessible. Going through the top to the back there are two slivery triangles, one on either side of the edge near the bottom. The corners near the bottom are covered by going through the right and bottom or the right side without going through the bottom. However, this coverage does not extend to the middle of the bottom of the back, even though coverage was needed all the way across, as the missing area is based on the tangent of the circle whose radius went directly to the back (tangent to the bottom edge).

                   Bt

                   Bk     Bk              Bk      Bk  Bt 

          Bt   L   T   R  Bt              L   T   R

 

                       

Since the attachment at a vertex leaves some surface area inaccessible as does a mid-edge mount, the best place is in the middle of a face.