How did Maxwell develop the magnetic field without relativity? Was it purely experimental? I don't see how else he would have developed any understanding for the magnetic field.
A long time before Maxwell wrote down a unified theory, Oersted discovered a connection between electricity and magnetism. In the development of electromagnetism, there were many bits and pieces of partial knowledge that were discovered and formulated by many different scientists. The popular ones (after whom we've named partial "laws") that immediately come to mind are Faraday (who came up with the concept of a field), Ampere, Gauss, Coulomb, etc. In fact, Faraday put together electric and magentic fields to make motors and generators. If I recall the history correctly, Maxwell understood the connections and put together all these partial results into a coherent mathematical formulation.
As it happened, it was noticed that this theory was not invariant under Galiean transformations and the challenge was to figure out which of the two concepts to be modified. To make a long story short, there were many proposals and notable the Michelson-Morley experiment, but the one that we now know to be correct was Einstein's theory of Special Relativity.
Some references for the history of E&M