When I try this, I don't see a LOT of Gibbs ringing. Depending on your purposes for the image, you could partially suppress this effect by post-filtering the image with something like

3dWinsor -irad 3.5 -prefix anatW anat+orig.HEAD

for example. To get a quick feel for if this might help, you can open the 'Disp' control panel in an AFNI image viewer, and then under the 'Tran 2D' menu, choose 'Winsor21' (which does a 2D Winsorizing filter). You can also open a RowGraph (1 is usually good) to plot the intensity values across the image (at the crosshairs) to see the effects of the filters, switching between datasets, etc.

It is possible to alter the amount of tapering used in the wsinc5 interpolation, but this makes very little difference in the result.

Finally, I'll note that 'resolution' isn't increased by any interpolation method, just the matrix on which the image/volume is reconstructed.