Motion is a typical reaction among animals and humans trying to reach better conditions in a changing world. This aspect has been studied intensively in social dilemmas where competing players' individual and collective interests are in conflict. Starting from the traditional public goods game model, where players are locally fixed and unconditional cooperators or defectors are present, we introduce two additional strategies through which agents can change their positions of dependence on the local cooperation level. More importantly, these so-called sophisticated players should bear an extra cost to maintain their permanent capacity to evaluate their neighborhood and react accordingly. Hence, four strategies compete, and the most successful one can be imitated by its neighbors. Crucially, the introduction of costly movement has a highly biased consequence on the competing main strategies. In the majority of parameter space, it is harmful to defectors and provides a significantly higher cooperation level when the population is rare. At an intermediate population density, which would be otherwise optimal for a system of immobile players, the presence of mobile actors could be detrimental if the interaction pattern changes slightly, thereby blocking the optimal percolation of information flow. In this parameter space, sophisticated cooperators can also show the co-called Moor effect by first avoiding the harmful vicinity of defectors; they subsequently transform into an immobile cooperator state. Hence, paradoxically, the additional cost of movement could be advantageous to reach a higher general income, especially for a rare population when subgroups would be isolated otherwise.