Beam-type periodic materials with high connectivity have displayed unique band gap behaviors analogous to locally resonant band gaps in acoustic metamaterials. In this study, structurally square re-entrant honeycomb, one highly connected lattice configuration featuring eight folded beams connected at each joint, is introduced to be the host structure of a smart material to tailor the elastic wave propagation. Finite length piezoelectric patches connected with negative capacitance shunting circuits are arranged on the beam surfaces, providing active adjustment via altering the parameters of shunting circuits. The characteristics of band structure of this smart structured material are investigated through the application of finite element method in conjunction with the Bloch theorem. Results demonstrate that the variation of internally resonant band gaps induced by the alteration of the piezoelectric patches to those positions and mechanical properties, can be precisely estimated by simple heuristic models proposed according to deformation characteristics of standing wave modes. This founding could promote the practical implementation of the highly connected honeycombs in the adaptive control to elastic wave.