You can add the harmonies the same way you added vocals.
...or am I trying to do something that is not possible?
There's really no limit to the number of times that you can add to an audio file. You do have to be a bit careful with the levels (to prevent clipping/distortion), because the volume increases every time you add sound. But, as long as you normalize or adjust the volume for 0dB peaks or less before rendering (exporting) you'll have no problems.
You can always
add (mix). But, you can't usually
subtract (un-mix). Except for example, if you have the original unmixed vocal track, and you haven't adjust the level or otherwise altered it, you
can invert it and mix it back-in to cancel/subtract it out. Of course if you have the original unmixed tracks, there's no reason to do that.
You might want to consider using more-advanced software... A DAW (Digital Audio Workstation) application would allow you to record many different tracks (at different times if you wish), and then mix all of the tracks at once, while adjusting the various levels (and other optional effects). If you try to add one track/layer at a time, you might get to the end and wish that the instruments or vocals were a bit louder or softer, and at that point you can only adjust the new track, or the already-mixed parts as a whole. I don't know of a free DAW, but
REAPER is only $60 USD for hobby or small-business use.