M.ted, I am into barbershop harmony. The old-timers spontaneously harmonizing is called "woodshedding". It can be done by 2 kinds of folks: those who have spent a lot of time singing a harmony part, and those who have unusual native talent. I'm finding after a few years of singing barbershop according to formal written arrangements, I'm finally starting to "hear" where to spontaneously sing baritone parts which had been mysterious to me before. But I've yet to become a woodshedder, which may be a lost art soon because so few people sing, and fewer have time to pursue this arrangement-less craft.
At any rate if you want to get that skill, all I can tell you is abundant practice in one part seems to be the key. (If you really want to get immersed in woodshedding you could start at the SPEBSQSA website.)
I think maybe this is why those close-harmony family groups can get so good, it's not just that their timbres blend but the fact that they simply have, or make, the luxury of spending a lot of time harmonizing, hours at a time, years on end. It's frustrating for the rest of us, we want harmony singing to be as easy as solo singing or chord-strumming, but it's more like piano and you have to have a bunch of people agree to play with you.