Adaptive filters are widely used in acoustic feedback cancellation systems and have evolved to be state-of-the-art. One major challenge remaining is that the adaptive filter estimates are biased due to the nonzero correlation between the loudspeaker signals and the signals entering the audio system. In many cases, this bias problem causes the cancellation system to fail. The traditional probe noise approach, where a noise signal is added to the loudspeaker signal can, in theory, prevent the bias. However, in practice, the probe noise level must often be so high that the noise is clearly audible and annoying; this makes the traditional probe noise approach less useful in practical applications. In this work, we explain theoretically the decreased convergence rate when using low-level probe noise in the traditional approach, before we propose and study analytically two new probe noise approaches utilizing a combination of specifically designed probe noise signals and probe noise enhancement. Despite using low-level and inaudible probe noise signals, both approaches significantly improve the convergence behavior of the cancellation system compared to the traditional probe noise approach. This makes the proposed approaches much more attractive in practical applications. We demonstrate this through a simulation experiment with audio signals in a hearing aid acoustic feedback cancellation system, where the convergence rate is improved by as much as a factor of 10.