Thanks Michael,
So that is why Karstens sweep worked for the wine bottle lable, while I could not do it using unwrap/Planar/Flow.

After applying Planar to the ConeUnwrap, the flat surface has Iso's going at an angle, which is no good for FLOW.
But the ConeUnwrap can be used, for the Sweep. First trim off the two straight sides. and use the straight side to do a sweep, with the two arcs as rails,
(which I think is equivalent to Karsten's sweep). (Then use FLOW. I have yet to try it.)
(I spent an hour doing it wrong this morning:-)

- Brian

(The bucket frustum has only one (outer surface) seam, I think.)

EDIT:
FLOW works well!
Select the two outer edges of the roundish cutouts on the cone/frustum, as Objects to Flow.
Select the surface of the frustum near Base circle, near seam, as Base surface.
Select flat swept surface of unwrap, near longer arc, near left straight edge, as Target surface.