"I am trying to improve my ability to hear/improvise three-part harmonies that don't share many notes."

I have started doing this more and more at church. I think the best thing for you to do is to play recordings and then try singing harmonies along with them. If a note doesn't sound good, stop singing it and try again next time. Gospel singers do this all the time.

The hard thing is writing down what you did. If you wish to do that, it is a whole nother ball game.

If a song is available in four parts (a hymn, for example) then you can take the tenor part and modify it some to make a nice descant.