It would concentrate the flux between the facing poles by providing a shorter, more permeable path between the free poles. With a less sensitive sensor this would mean needing a smaller angle to reach the saturation extremes, and therefore a more linear response though on a smaller range of movement. However, if you already get the proper signal swing for the proper angle adding a flux ring would be counter-productive. The other advantage to a flux ring is that it will shield the free poles, reducing the stray magnetic field, and reducing the possibility that one sensor will affect the reading of another nearby. It will also make it harder for it to pick up stray iron filings... your workspace looks very tidy, but in my case that's a realistic problem.