I present a novel mechanism to boost magnetic field amplification of newly born neutron stars in core collapse supernovae. In this mechanism, that operates in the jittering jets explosion mechanism and comes on top of the regular magnetic field amplification by turbulence, the accretion of stochastic angular momentum in core collapse supernovae forms a neutron star with strong initial magnetic fields but with a slow rotation. The varying angular momentum of the accreted gas, which is unique to the jittering jets explosion mechanism, exerts a varying azimuthal shear on the magnetic fields of the accreted mass near the surface of the neutron star. This, I argue, can form an amplifying effect which I term the stochastic omega (Sω) effect. In the common αω dynamo the rotation has constant direction and value, and hence supplies a constant azimuthal shear, while the convection has a stochastic behavior. In the Sω dynamo the stochastic angular momentum is different from turbulence in that it operates on a large scale, and it is different from a regular rotational shear in being stochastic. The basic assumption is that because of the varying direction of the angular momentum axis from one accretion episode to the next, the rotational flow of an accretion episode stretches the magnetic fields that were amplified in the previous episode. I estimate the amplification factor of the Sω dynamo alone to be ≈ 10. I speculate that the Sω effect accounts for a recent finding that many neutron stars are born with strong magnetic fields.