Hello Scott,

Your description (slender, white, resembling a Teramachia or Turrid)
sounds very much like the genus Benthovoluta which, in spite of the
name, is not a volute but a member of Turbinellidae.  I never heard of
the species rinkinseni, but if you have access to Barry Wilson's
Australian Marine Shells, check out Volume 2, Plate 8 - Benthovoluta
claydoni, and see if that's what you have.

Regards,
Paul Monfils