If there are original refrains, codas, or bridges, you’re probably singing contemporary songs with old words. Here’s one of the most well-known examples from Chris Tomlin. He uses several of John Newton’s stanzas, but adds the “My chains are gone” refrain.
http://www.youtube.com/watch?v=HkAZteruVAw
This sort of thing is not completely unheard of in hymnody, particularly in gospel hymnody. (Think of how “Marching to Zion” adds a refrain to Dr. Watts’ text.) But Tomlin’s version turns the strophic form into a contemporary composition, giving it that emotional high-point (in which he hits a high G, no less) that sells so well in commercial recordings. Additionally, if you look at the trajectory of Newton’s text, the “My chains are gone” motif is unnecessarily awkward.
And, if you’re going to sing “Amazing Grace,” why not just sing “Amazing Grace?”