I'm less experienced than you, but I'm okay at coming up with independent parts that sound good together. The main trick for me is paying attention to "density" of music, which is a combination of tempo, number of voices, amount of dissonance, frequency of chord changes, etc. Think of it as a unified measurement of how much pressure you're putting on the listener. I'm convinced that almost any song can be made better by tweaking density up or down, and the different ways of tweaking it are almost interchangeable. To make something denser, you speed it up or add a voice or change the chord. If you overshoot, you can then reduce density in some other way.

So coming back to the topic, when I want to add a new voice, first I play the section on repeat and pay attention to density. If it's "full", I either leave it alone or dial down the density to make room, kill some of my darlings. That's the hard part. And when there's room, you can add almost any half decent melody and it will work, as long as you don't overshoot. Simple as that.

Sorry about the rambling - I'm writing this as much for myself as for you...