This paper introduces a high-performance hybrid algorithm, called Hybrid Hypervolume Maximization Algorithm (H2MA), for multi-objective optimization that alternates between exploring the decision space and exploiting the already obtained non-dominated solutions. The proposal is centered on maximizing the hypervolume indicator, thus converting the multi-objective problem into a single-objective one. The exploitation employs gradient-based methods, but considering a single candidate efficient solution at a time, to overcome limitations associated with population-based approaches and also to allow an easy control of the number of solutions provided. There is an interchange between two steps. The first step is a deterministic local exploration, endowed with an automatic procedure to detect stagnation. When stagnation is detected, the search is switched to a second step characterized by a stochastic global exploration using an evolutionary algorithm. Using five ZDT benchmarks with 30 variables, the performance of the new algorithm is compared to state-of-the-art algorithms for multi-objective optimization, more specifically NSGA-II, SPEA2, and SMS-EMOA. The solutions found by the H2MA guide to higher hypervolume and smaller distance to the true Pareto frontier with significantly less function evaluations, even when the gradient is estimated numerically. Furthermore, although only continuous decision spaces have been considered here, discrete decision spaces could also have been treated, replacing gradient-based search by hill-climbing. Finally, a thorough explanation is provided to support the expressive gain in performance that was achieved.