In fact every magnetic field has is produced by electric currents of some sort and every electric field gives rise to an electric current which produces a magnetic field with direction given by the Right Hand Grip Rule.
It is in fact impossible to produce an electric field without producing a current, however tiny, which gives rise to a magnetic field, and the presence of a magnetic field indicates there must be a current of some sort. In fact the magnetic field produced by a magnetic is due to electrons circling atoms. We can think of the electrons as constituting tiny currents which produce a significant magnetic field when the currents line up in the same direction.
When the currents align in the same direction,the magnetic fields add in the direction through the centre of the current loops and cancel in other directions, producing a significant magnetic field.