Efficient stimulation of deeply located brain regions with transcranial magnetic stimulation (TMS) poses many challenges, arising from the fact that the induced field decays rapidly and becomes less focal with depth. We propose a new method to improve the efficiency of TMS of deep brain regions that combines high permeability cores, to increase focality and field intensity, with a coil specifically designed to induce a field that decays slowly with increasing depth. The performance of the proposed design was investigated using the finite element method to determine the total electric field induced by this coil/core arrangement on a realistically shaped homogeneous head model. The calculations show that the inclusion of the cores increases the field's magnitude by as much as 25% while also decreasing the field's decay with depth along specific directions. The focality, as measured by the area where the field's norm is greater than of its maximum value, is also improved by as much as 15% with some core arrangements. The coil's inductance is not significantly increased by the cores. These results show that the presence of the cores might make this specially designed coil even more suited for the effective stimulation of deep brain regions.