To add to it; A third above and a sixth below will give the same notes but an octave apart. A danger with a harmony line above the melody in pitch is that the melody line can get lost because we tend to hear the top line, so below is better, at least to start with. A third below will also work but, again bear the point made by GSS about avoiding continuous parallel motion and try to work in some contrary motion. You also need to bear in mind the range of the instruments you are writing for.
If you do want a harmony line above the main melody, then you need to give it to an instrument capable of playing quietly or tell the singer to sing quietly or only use it for a limited time. In hymn singing a descant is a harmony line pitched above the main melody and I am sure you will have noticed that it is usually used for just one verse as it provides some variety towards the end of a number of musically identical verses.