Antipodal distance-transitive covers with primitive quotient of diameter two

As a consequence of a famous theorem by Derek Smith, an unknown distance-transitive graph is either primitive of diameter at least two and valency at least three or is antipodal, bipartite, or both. In the imprimitive cases an unknown graph must have a primitive core of diameter at least two and valency at least three. It seems that the known list of primitive graphs is complete. Here, starting from an earlier work by Brouwer and Van Bon, we find every distance-transitive antipodal cover whose primitive quotient is one of the known distance-transitive graphs of diameter two and valency at least three.