At the end, when the elusive caterpillar that Dinky and Boomer chase throughout the film turns into a butterfly, it's a reflection on the larger plot where the hunters and prey reconcile.
When Copper starts leaving his barrel to go play with Todd, Chief makes a point to tell him how upset Amos will be about it. True to this prediction, Copper is put on a leash for his misbehavior. Before Copper started wandering off, what was the biggest difference in how Amos treated Chief and himself? Chief was always kept on a leash. He had been trying to warn Copper about losing his privileges the same way that he did when he was younger.
Although Tod was Demoted to Extra in the second film and Copper was shown to shine in the film, if you think about who was the protagonist and the most important character in the first film, you'll realize that the focus merely switched over from Tod to Copper.
First film: Focused on Tod
Second film: Focused on Copper
Why did Amos keep Chief on a leash but let Copper go unrestrained? It's all well and good to teach your dog to stay near your house on its own accord while it's still young, but shouldn't Amos have been personally watching him during this learning period? There are wild animals all around their house! A puppy Copper's size could get eaten if it wanders too far.
The Fox and the Hound begins with the baby fox's mother hiding him and sacrificing her own life for his, Bambi-style. However, foxes almost never have a single cub. This mother wasn't just carrying her baby to safety. She was carrying to safety the only baby of the litter who survived the hunt.