Based on the quantum-assisted genetic algorithm (QAGA) [11] and related approaches we introduce several modifications of QAGA to search for more promising solvers on (at least) graph coloring problems, knapsack problems, Boolean satisfiability problems, and an equal combination of these three. We empirically test the efficiency of these algorithmic changes on a purely classical version of the algorithm (simulated-annealing-assisted genetic algorithm, SAGA) and verify the benefit of selected modifications when using quantum annealing hardware. Our results point towards an inherent benefit of a simpler and more flexible algorithm design.