Greetings,
Having wrestled with this issue myself, let me ask if both (or either) of you have a check next to "Stretch columns to fit window"? If so, please try clicking on that option to remove the checkmark. Likewise, moving your cursor into the column headings and hovering over the division between which ever headings may be present, please see if you can move the divisions to the left (even if this action makes the columns you have look a bit 'wonky' for the moment). When you have finished, the right hand side of Clementine's window should be completely blank.
At this juncture, Right-click on any column header to open the drop down menu you both are already familiar with. Select whichever column header you wish to add (just one for now, please) and it ought to appear relative to its position in the drop down menu to your existing columns (i.e., before, between or after them). Please note that this will require a bit of experimentation and exploration on your parts to get the columns to appear as you wish them to. As far as I have been able to tell, one isn't able to change the order in which the columns appear, only whether or not they appear at all.
I hope this information is helpful to you.