I actually figured out how to control the order in which the buttons located below the "Add-on Tools" are listed.
There is an optional parameter for app.addToolButton called nPos: that can be used. I believe the lower the number used, the higher up on the list the button will be displayed. It sort of worked. The list of buttons did re-arrange themselves but not exactly the way I wanted.
Probably all I had to do was to rename the JS files with numbers "1" through "13" and the buttons with be listed correctly.
There is also a nPos parameter that can be used to order the toolbar buttons.