Context. Observations of dwarf galaxies suggest the presence of large-scale magnetic fields. However the size and slow rotation of these galaxies appear insufficient to support a mean-field dynamo action to excite such fields. Aims. Here we suggest a new mechanism to explain large-scale magnetic fields in galaxies that are too small to support mean-field dynamo action. The key idea is that we do not identify large-scale and mean magnetic fields. In our scenario the magnetic structures originate from a small-scale dynamo which produces small-scale magnetic field in the galactic disc and a galactic wind that transports this field into the galactic halo where the large turbulent diffusion increases the scale and order of the field. As a result, the magnetic field becomes large-scale; however its mean value remains vanishing in a strict sense. Methods. We verify the idea by numerical modelling of two distinct simplified configurations, a thin disc model using the no-z approximation, and an axisymmetric model using cylindrical r,z coordinates. Results. Each of these allows reduction of the problem to two spatial dimensions. Taken together, the models support the proposition that the general trends will persist in a fully 3D model. We demonstrate that a pronounced large-scale pattern can develop in the galactic halo for a wide choice of the dynamo governing parameters. Conclusions. We believe that our mechanism can be relevant to explaining the presence of the fields observed in the halos of dwarf galaxies, and maybe elsewhere. We emphasize that detailed modelling of the proposed scenario needs 3D simulations, and adjustment to the specific dynamo governing parameters of dwarf galaxies.