I suggest you take the other way around approach.
Instead of extending the existing enumeration, create a larger one and create a subset of it. For example, if you had an enumeration called PET and you wanted to extend it to ANIMAL you should do this instead:
public enum ANIMAL {
WOLF,CAT, DOG
}
EnumSet<ANIMAL> pets = EnumSet.of(ANIMAL.CAT, ANIMAL.DOG);
Be careful, pets are not immutable collections, you might want to use Guava or Java9 for more safety.