We approach the problem of bulk acceleration in relativistic, cold, magnetized outflows, by solving the momentum equation along the flow, a.k.a. the wind equation, under the assumptions of steady-state and axisymmetry. The bulk Lorentz factor of the flow depends on the geometry of the field/streamlines and by extension, on the form of the "bunching function" S=r^2 B_p/ A, where r is the cylindrical distance, B_p the poloidal magnetic field, and A the magnetic flux function. We investigate the general characteristics of the S function and how its choice affects the terminal Lorentz factor gamma_f and the acceleration efficiency gamma_f/mu, where mu is the total energy to mass flux ratio (which equals the maximum possible Lorentz factor of the outflow). Various fast-rise, slow-decay examples are selected for S, each one with a corresponding field/streamlines geometry, with a global maximum near the fast magnetosonic critical point, as required from the regularity condition. As it is proved, proper choices of S can lead to efficiencies greater than 50%. Last, we apply our results to the momentum equation across the flow, in an effort to estimate their validity, as well as identifying the factors that lead to an accurate full-problem solution. The results of this work, depending on the choices of the flow integral mu, can be applied to relativistic GRB or AGN jets.