That way does work well.

I do it this way, I compress the 2nd and 3rd separately from the first

The 2nd and third is the whipped cream, the lead is the cheery on top. Get the picture. A large one too.

I don'thave enough info on your approach but when I do these kind of vocals generally every back up is quadrupled. Depending on the part I'll add quite a bit of top (10k-15K)and cut some bottom and low mids. I'll use anywhere from 2-6db of compression w/slow attack, med to fast release. Very little reverb but maybe some kind of pitch/double. Sometimes I'll pan each of the quads hard left/hard right then 11:00/2:00. This might change depending on how the parts interact. This approach has always resulted in real creamy & present back ups. The lack of reverb really helps with the presence. Of course you could spend a couple of hours eloborating on the subject this is just one approach.
If you only have one track of each part and you work in a DAW you could copy each track a few times and move each one a couple of ms. Not as cool as indv parts but still cool.