This paper focuses on distributed signal estimation in topology-unconstrained wireless acoustic sensor networks (WASNs) where sensor nodes only transmit fused versions of their local sensor signals. For this task, the topology-independent (TI) distributed adaptive node-specific signal estimation (DANSE) algorithm (TI-DANSE) has previously been proposed. It converges towards the centralized signal estimation solution in non-fully connected and time-varying network topologies. However, the applicability of TI-DANSE in real-world scenarios is limited due to its slow convergence. The latter results from the fact that, in TI-DANSE, nodes only have access to the in-network sum of all fused signals in the WASN. We address this low convergence speed by introducing an improved TI-DANSE algorithm, referred to as TI-DANSE+, in which updating nodes separately use the partial in-network sums of fused signals coming from each of their neighbors. Nodes can maximize the number of available degrees of freedom in their local optimization problem, leading to faster convergence. This is further exploited by combining TI-DANSE+ with a tree-pruning strategy that maximizes the number of neighbors at the updating node. In fully connected WASNs, TI-DANSE+ converges as fast as the original DANSE algorithm (the latter only defined for fully connected WASNs) while using peer-to-peer data transmission instead of broadcasting and thus saving communication bandwidth. If link failures occur, the convergence of TI-DANSE+ towards the centralized solution is preserved without any change in its formulation. Altogether, the proposed TI-DANSE+ algorithm can be viewed as an all-round alternative to DANSE and TI-DANSE which (i) merges the advantages of both, (ii) reconciliates their differences into a single formulation, and (iii) shows advantages of its own in terms of communication bandwidth usage.