I was thinking that but dismissed it since that would put the edges pointed towards the typical push from the hands. I'm thinking this would quickly fold the edges up and make a mess with the foam "cork" tape we typically use these days.

Starting the foam cork from the tops neatly isn't a big deal. I already do this when starting at the base anyway. I cut a wedge off and then slice the thick center along the cut side of the wedge to a thining taper. The tape starts a little in from the desired end and in this case would turn towards the stem for the first turn or turn and a half and then come back to the desired bias angle for the run down to the other end. There's going to be a transition where the edge comes back on itself but with a little trial and error the cross over will be under the bars where you don't see it. Up top there would only be a nice looking tape edge. This is the normal method of course but slicing the tape thinner really helped me get a nice clean looking start.