The experimental fact behind is the Oersted observation that a compass needle is deflected by a conducting wire. Before the invention of the battery by Volta, it was not possible to have a continuous current in a wire.
If the wire is aligned north-south, (that is aligned with the needle), after closing the circuit the needle deflects from the north, and the angle of deflection depends on the intensity and polarization of the current.
Because compass alignment is a magnetic phenomena, the dominant theory was to postulate a magnetic field created by the current. And the magnetic field should be transverse to the wire to explain what happens. The cross product fits well, because the deflection angle changes direction (east or west) if the wire is above or behind the compass, for the same polarization.
It is worth to remark that the French mathematician Ampère notices, very soon afterwards, that two conduction wires were attracted or repelled depending on the direction, intensity and polarization of the current. He developed a quantitative theory to calculate that forces. And postulates that magnets were only the side effect of microscopic currents. So, the compass deflection effect was for him basically a force between currents. There was no such a thing as a magnetic field.
But the existence of that invisible current was probably considered too speculative. And Faraday idea of using small iron pieces to show the magnetic field lines was a good argument for the existence of a magnetic field.