When a BIBD or a Group Divisible design with $\lambda_2 = \lambda_1 + 1$ is extended by certain disjoint and binary blocks the resulting structure is proved $E$-optimal. A BIBD abridged by a certain number of such blocks is also shown $E$-optimal. These optimality results hold over the class of all block designs (with the respective sets of parameters). Proofs rely mainly on averaging information matrices, which proves useful in many settings related to design optimality.