It may be worth looking into the presets of any reverb or delay FX plugins you have to see if there is a "doubling" preset. Fiddle from there, if there is, and see what about that effect is effective.

Doubling and hard-panning the tracks WITH the slight judicious offset mentioned will make it sound like there is a track center even if there isn't. I'm not sure treating them differently with other effects eliminates or accentuates that mirage. If it does, maybe three virtual tracks is an answer?
If you sing it again for a doubling effect, remember that you don't have to use it end-to-end. Doubling important words, phrases, or sections may be enough to get the desired results. Same is true for any other method, I reckon.