As it sounds like you have a good feel for the right hand, you might lean on that for some insight into the left. Add some harmony on the right, and then poke around among the likely suspects on the left, to see who has those notes. Alternatively, you could just ignore your right hand for a while and learn the left like you did the right, just play some stuff over there for your amusement and it will start to catch on.
If the written material you're working with isn't anything like the tunes you play by ear, of course it might be interesting to dig some of that stuff up, likely easy to find online.
You may have noticed a fairly standard pattern, that moves around the left side depending on the key of your tune - three major chords, with the root in the middle, three minor chords above them. Depending on the tune - the last of those three minors may actually be a major / dominant 7. If you try the dominant 7 chord with your major chords, it will work better with one of them, and it will always be the same one (as long as the tune remains solidly in one key.) Don't bother with the diminished chords. This is in the same vein as the "three chord trick", I've just never cared for that term because 1) if I could stand to play anything with only three chords, they'd probably include at least one minor chord, and 2) "trick" sounds like there might be something going on other than just playing the chords that are called for.