This paper extends the funnelling behavior to offer a low-cost flexible guidance of mobile entities towards a circular region goal with the guarantee of enforcing an orientation within a predefined tolerance interval. The key requirements are the same as the tunnelling control, i.e. a low and constant cost update of the control even when the goal parameters change (distance and relative orientation of the goal, position tolerance radius, orientation tolerance interval, desired speed). The smoothness and the optimality of the resulting trajectory being of high importance the paper qualitatively compares the trajectories produced by both tunnelling algorithms. The new relaxed approach appears to produce smoother and shorter path for path made of a succession of large region goals. These qualities and its low cost advocate for its exploitation for moving through large dynamically changing regions without precise a priori planning.