We compare the nonlinear magneto-optical rotation (NMOR) effect in a single-beam probe scheme and a two-beam Raman-coherence-assisted scheme using cold rubidium atoms produced with an integrating sphere apparatus. In both schemes, NMOR arising from D1 and D2 excitation combinations was studied. We show that adding a Raman-coherence manipulation laser can substantially increase the NMOR, and the largest NMOR enhancement occurs when both probe and Raman-coherence lasers couple the same D2 transitions with different detunings. For far-off resonance excitation with a weak probe laser, we show that the Raman-coherence-assisted NMOR enhancement can be about 6 times larger than the single probe scheme operated under the same conditions. The Raman-coherence-enhanced NMOR is also shown to be substantially larger than that from an optically pumped magnetometer using a single strong circularly polarized probe.