Many real-life problems involve multiple conflicting objectives; hence, the decision maker is provided with a set of trade-off solutions, the Pareto front. While many methods to compute Pareto fronts have been proposed in the mathematical programming literature, comparatively few approaches are available for constraint programming (CP). One of the main state-of-the-art algorithms in CP is a branch-and-bound method that uses a Pareto global constraint, denoted here as MOBAB-CP. In this work, we adapt the SAUGMECON algorithm, a well-known and efficient ε-constraint method, in a CP solver. We also propose a new algorithm that combines SAUGMECON with the Pareto global constraint. Experimental results show that the proposed algorithm consistently achieves better results than our CP implementation of SAUGMECON and is competitive with MOBAB-CP, outperforming it on several of the studied problems.