This work proposes novel hybrid mixed-membership blockmodels (HMMB) that integrate three canonical network models to capture the characteristics of real-world interactions: community structure with mixed-membership, power-law-distributed node degrees, and sparsity. This hybrid model provides the capacity needed for realism, enabling control and inference on individual attributes of interest such as mixed-membership and popularity. A rigorous inference procedure is developed for estimating the parameters of this model through iterative Bayesian updates, with targeted initialization to improve identifiability. For the estimation of mixed-membership parameters, the Cramer-Rao bound is derived by quantifying the information content in terms of the Fisher information matrix. The effectiveness of the proposed inference is demonstrated in simulations where the estimates achieve covariances close to the Cramer-Rao bound while maintaining good truth coverage. We illustrate the utility of the proposed model and inference procedure in the application of detecting a community from a few cue nodes, where success depends on accurately estimating the mixed-memberships. Performance evaluations on both simulated and real-world data show that inference with HMMB is able to recover mixed-memberships in the presence of challenging community overlap, leading to significantly improved detection performance over algorithms based on network modularity and simpler models.