Some "schooling" fish school a lot more than others. In my experience, neon tetras never really school much unless they are really scared of something. Of all the tetras, I think rummynose are by far the best schoolers.
While you don't want to "scare" your fish into schooling, sometimes adding one, larger fish to your tank will promote schooling a bit more. This is assuming you have the space. But if your tank is a 20 long or larger, maybe consider adding one (regular, not dwarf) gourami like a Pearl Gourami or a Gold Gourami. Or maybe a single betta (female might be better than a male just to prevent any fin nipping by the tetras).
But as to your idea, I doubt adding just a few more neons will result in much of a change. It might help (a tiny bit), but don't expect any miraculous improvements.