Metaheuristics such as genetic algorithms and local search traditionally explore a solution space to solve combinatorial optimisation problems such as scheduling or packing problems. Hyper-heuristics explore a heuristic space rather than the solution space directly and were introduced to overcome some of the challenges associated with searching the solution space. Usually, only one space, i.e., the solution or program, is explored to solve the problem at hand. More recently, there have been initiatives to explore both the solution and heuristic space at the same time. This has been referred to as bi-space search. In previous work, the effectiveness of bi-space search, using iterated local search, for one-dimensional bin packing was shown. This study evaluates the hypothesis that a bi-space search using genetic algorithms to explore the solution and heuristic spaces, and a single point search selection perturbative hyper-heuristic to optimise when to switch between spaces (SPHH), will result in performance improvements over bi-space searches in previous work for bin packing. The proposed approach is evaluated for one-dimensional (1D) and two-dimensional (2D) bin packing problems. The study found that the SPHH performed better than searching each of the spaces separately for both 1D and 2D bin packing problem instances. Furthermore, the proposed approach outperformed the state-of-the-art bi-space searches for bin packing.