For the cranberry relish: Combine the cranberries, sugar, orange zest, salt and 1 cup water in a medium saucepan and bring to a simmer over medium-high heat. Simmer, stirring occasionally, until the cranberries break down and the relish has thickened, about 12 minutes. Let cool completely. The relish can be made up to 3 days ahead and kept refrigerated in an airtight container.
For the mini cornbread muffins: Preheat the oven to 350 degrees F. Spray a 24-cup mini-muffin tin with cooking spray.
Whisk together the cornmeal, flour, sugar, baking powder and salt in a large bowl. In another bowl, whisk together the milk, egg and butter until well combined. Pour the milk mixture into the cornmeal mixture and whisk until just incorporated. Spoon the batter evenly into the prepared muffin cups. Bake until a wooden pick inserted into the center of a muffin comes out clean, about 12 minutes. Let cool in the tin 5 minutes. Remove from the tin.
For the cranberry cornbread bites: Stir together the butter and chives in a small bowl. Cut the muffins in half crosswise with a serrated knife. Spread a little of the chive butter on both sides of a muffin. Top the bottom half with a piece of ham. Spoon a dollop of cranberry sauce on top. Sandwich with the muffin top and place on a platter. Repeat with the remaining muffins.