Owing to the unique wave retarding feature, Acoustic Black Hole (ABH) structures with a standard power-law thickness profile have been extensively investigated for various applications in structural vibration and sound radiation mitigation. As an attempt to achieve the best possible ABH effects under a prescribed minimum thickness, this paper reports an alternative ABH thickness profile. Employing the previously developed semi-analytical Daubechies wavelet plate model, a new ABH profile is obtained through a systematic optimization approach. The new profile features a thickness variation which is different from the standard one. Specifically, the thinnest part of the optimized profile is off-set from the indentation centre, thus forming a ring-shaped flexible area, which is shown to be conducive to energy focalization and dissipation with the deployment of a damping layer. The new profile promotes the formation of local (n-, 1) and (n-, 2) modes which entails damping increase, alongside a bi-directional wave trapping process, both within the ring-shaped area. As a result, a significant increase in the system damping and a reduction in the structural vibration and sound radiation are observed.