Is it common practice to have a separate "Further Discussion" after the conclusion in an academic article?
Virtually all papers I've seen end with the conclusions section (otherwise, it wouldn't be the conclusions!).

The best way to incorporate such changes would be to include it at the end of the current discussion section. Perhaps you can subtitle it "A brief note on X," where X is the topic of the comment. Alternatively, you could include it as an appendix if it requires a more extensive and extended discussion.