This paper addresses the challenge of topology-independent (TI) distributed adaptive node-specific signal estimation (DANSE) in wireless acoustic sensor networks (WASNs) where sensor nodes exchange only fused versions of their local signals. An algorithm named TI-DANSE has previously been presented to handle non-fully connected WASNs. However, its slow iterative convergence towards the optimal solution limits its applicability. To address this, we propose in this paper the TI-DANSE+ algorithm. At each iteration in TI-DANSE+, the node set to update its local parameters is allowed to exploit each individual partial in-network sums transmitted by its neighbors in its local estimation problem, increasing the available degrees of freedom and accelerating convergence with respect to TI-DANSE. Additionally, a tree-pruning strategy is proposed to further increase convergence speed. TI-DANSE+ converges as fast as the DANSE algorithm in fully connected WASNs while reducing transmit power usage. The convergence properties of TI-DANSE+ are demonstrated in numerical simulations.