The highly aligned cortical microtubule array of interphase plant cells is a key regulator of anisotropic cell expansion. Recent computational and analytical work has shown that the non-equilibrium self-organization of this structure can be understood on the basis of experimentally observed collisional interactions between dynamic microtubules attached to the plasma membrane. Most of these approaches assumed that new microtubules are homogeneously and isotropically nucleated on the cortical surface. Experimental evidence, however, shows that nucleation mostly occurs from other microtubules and under specific relative angles. Here, we investigate the impact of directed microtubule-bound nucleations on the alignment process using computer simulations. The results show that microtubule-bound nucleations can increase the degree of alignment achieved, decrease the timescale of the ordering process and widen the regime of dynamic parameters for which the system can self-organize. We establish that the major determinant of this effect is the degree of co-alignment of the nucleations with the parent microtubule. The specific role of sideways branching nucleations appears to allow stronger alignment while maintaining a measure of overall spatial homogeneity. Finally, we investigate the suggestion that observed persistent rotation of microtubule domains can be explained through a handedness bias in microtubule-bound nucleations, showing that this is possible only for an extreme bias and over a limited range of parameters.