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We recently demonstrated that hexagonal EuIn₂As₂ exhibits a zero-field low-symmetry helical antiferromagnetic order that makes the compound a stoichiometric magnetic topological-crystalline axion insulator[1]. It is predicted that the topological properties of the surface states can be tuned according to the direction and strength of an applied magnetic field. We continue our investigation of EuIn₂As₂ by performing single-crystal neutron diffraction experiments while using an external magnetic field in order to determine field-induced changes to the magnetic order. With increasing field applied in the basal plane, our results are consistent with the spins first flopping out of the plane while keeping a complex in-plane spiral periodicity. The out of plane component is then progressively suppressed and a coplanar fan-type structure appears before the fully field polarized state prevails. The intermediate-field magnetic ordering may give rise to exotic topology associated with a non-coplanar spin texture.