From the previous post: Ten Tips for Writing a Song, I discovered the existence of the 'half-rhyme'.
Doing a bit of research, here's what Wikipedia says. It is the use of words ending with the same consonants, such as 'moon' and 'on', 'soul' and 'all'.
I also discovered yet another kind of rhyming called the eye rhyme. These involve using words that have similar spellings (but are pronounced differently) as if they actually rhymed. Laughter and Slaughter were used as an example.
Another related technique I discovered is assonance. This works with the repeated and strategic use of vowels to create a kind of rhythm in the phrases that gives the lyrics an attractive structure.