Marion, you've raised a really good question! I'll provide a perspective of someone who's pretty good at developing chords by ear, but who gets stumped from time to time. I have three suggestions.

(1) Know your genre! There are clearly different chords for different music styles - Celtic, Ragtime, Blues. If, for example, you haven't encountered Eastern European dance music be prepared to throw your standard chord progressions out the window. But once you get a bit of experience the task is a lot easier.

(2) Examine the most forceful melody note in the difficult phrase, probably the first note. Then look for chords which contain that note. If you're in the key of C start with the simplest chords that you may expect to find and if necessary then move on to the Major, Minor or Seventh chords. Hence if you find an Ab in the melody you'll find it in the F Minor chord. Only then move on to Chords which are outside of what you would normally find, say Ad or Db. In my experience the third step is rarely necessary or fruitful.

(3) Experiment with a program such as ABCMUS (Shareware). I can't recall whether the un-registered version calculates chords for you - I think it does. While the results can vary from spectacular to awful it can give you unexpected clues as to alternative chords. Apart from the 'strageness factor' which you should play around with, there are other complex algorithms to tweak - but I don't have the theory background to make much sense of it.

Last weekend I was asked to sing Nancy Spain. I refused on the basis that I wasn't comfortable with something about it. I sat down the follwing day and completely re-wrote the usual chords for it. There is no such thing as 'the right chords' because only you will know if you're satisfied with what you're playing. Enjoy the quest. Regards, John