Call me weird, but I've done somewhat ok attempting to avoid the oils/fines with a French press by basically putting a paper towel over the glass before putting the plunger thing back into it. So the paper towel wraps around the metal mesh, not only adding its filtering to the mesh but also improving the seal it makes against the side of the glass. It does mean that I have to press it down more slowly before pouring, but that's just because it's doing its job so I don't mind. It also means that the mesh/plunger bit requires less cleaning afterward.