Not everyone should overhead press a barbell. There are often postural restrictions, stability issues, or motor control deficits. But for those who are able to maintain a neutral spinal position, keep the rib cage from flaring, and have the mobility to press overhead without pain or dysfunction, then the false grip can be an extremely useful tool.
Your press should start from the rack in order to optimize the grip, shoulder position, and pillar stability before any dynamic pressing happens. Stabilize your grip and shoulders under the bar first, then walk back with control just as you would a heavy squat.
While maintaining neutral pillar positioning throughout the press, make sure you tap into the stable power of the irradiation effect by gripping all five fingers against the bar maximally. Engage on the way up, and maintain stability through accentuating the eccentric (slowly lowering) on the way down.
Now, to keep the bar from falling down and knocking you out, avoid push pressing or momentum-based overhead work when using the false grip.