Problems involving the large deformation of hyperelastic materials in contact are of interest in both industry and research. Due to the significant non-linearities they present, numerical tools such as finite element discretization, a quasi-static framework, and Newton-like solvers are commonly used to solve these problems. However, convergence difficulties can arise from bifurcations and severe non-linearities. Predictor-corrector or arc-length methods, which introduce a path-following constraint and treat the load increment as an additional unknown within the finite element space, can improve convergence. While these methods add computational complexity, we demonstrate that incorporating a continuation algorithm accelerates the overall iterative process. Following the work of Léger et al. (2015), we extend the Moore-Penrose continuation algorithm to contact mechanics in hyperelasticity. We demonstrate its implementation in an existing finite element code and its effectiveness in solving large deformation contact problems, including industrial examples in two and three dimensions. Furthermore, we show that the proposed algorithm is compatible with common contact-solving methods, such as penalization and augmented Lagrangian methods. Through additional contact mechanics examples, we demonstrate that achieving large deformations in hyperelastic materials—often associated with severe mesh distortion—is possible using the proposed continuation algorithm, an updated Lagrangian formulation, and an adaptive remeshing strategy.