We describe a theoretical investigation of a four-wave mixing (FWM) scheme in a six-level atomic system driven by a field with orbital angular momentum and making use of two electromagnetically induced transparency (EIT) control fields. The obtained results allow us to control the helical phase of the output FWM field by varying the intensities of the two EIT control fields as well as the detuning of the probe field. More interestingly, we can achieve FWM output fields that are very stable under variation of the system parameters and allow the helical phase twist to be manipulated and suppressed, if desired, by adjusting the probe detuning. In this way, our scheme may open a tool to control the helical phase in a compact, efficient, and robust way.