Peel the cooked eggs and cut them in half lengthwise.
Add the yolks to a medium bowl and place the whites aside.
Mash and stir the egg yolks along with the mayonnaise, ham crumbles, Dijon mustard, sweet relish, and black pepper. Taste. If lacking salt, add a tiny bit. Depending on how salty the ham is, you may or may not decide you need some.
Place the egg yolk mixture into a piping bag or a large (gallon-size) zip-top bag fitted with a ½" piping tip, or snip off a ½" corner from the bag.
Generously fill the egg white shells with the deviled ham and egg yolk mixture and place the stuffed eggs on a plate for serving.
Sprinkle the top of the eggs with chopped chives, and chill before serving.
Notes
Deviled ham and eggs can be stored in the refrigerator for up to 2 days if covered well.