A new family of IBM Hamiltonians, characterized by certain parameter values, was found about 15 years ago by Alhassid and Whelan to display almost regular dynamics, and yet these solutions to the IBM do not belong to any of the known dynamical symmetry limits (vibrational, rotational and gamma - unstable). Rather, they comprise an "Arc of Regularity" cutting through the interior of the symmetry triangle from U(5) to SU(3) where suddenly there is a decrease in chaoticity and a significant increase in regularity. A few years ago, the first set of nuclei lying along this arc was discovered. The purpose of the present work is to search more broadly in the nuclear chart at all nuclei from Z = 40 - 100 for other examples of such "regular" nuclei. Using a unique signature for such nuclei involving energy differences of certain excited states, we have identified an additional set of 12 nuclei lying near or along the arc. Some of these nuclei are known to have low-lying intruder states and therefore care must be taken, however, in judging their structure. The regularity exhibited by nuclei near the arc presumably reflects the validity or partial validity of some new, as yet unknown, quantum number describing these systems and giving the regularity found for them.