It has been demonstrated by McGregor that semi-Lagrangian advection techniques may be efficiently applied to a cubic gnomonic grid on the sphere. Despite the nonorthogonal nature of that grid, the accuracy is superior to that of conventional latitude–longitude grids. The present paper demonstrates even greater accuracy by applying similar techniques to the related conformal-cubic grid devised by Rančić et al.; an important new feature is a simple iterative technique for the inverse calculation of grid coordinates. Advection over the vertices of the grid exhibits none of the problems that occur over the poles of a latitude–longitude grid. A stretched grid configuration is also presented showing further improvements. It is finally shown that the departure points may be interpolated onto a B-grid version and advection performed simply on the staggered grid without loss of accuracy.