Spin-polarized scanning tunneling microscopy is used to identify the magnetic state of different thin films on a Re(0001) substrate, which becomes superconducting below 1.7 K. All magnetic films contain an Fe/Ir interface, which is known to facilitate the emergence of non-collinear magnetic order. For Fe monolayers on ultra-thin Ir films of different thickness we find several different atomic-scale magnetic states. For Pd/Fe bilayers we find nano-scale spin spirals as magnetic ground states for Ir thicknesses of three and four atomic layers. In applied magnetic fields skyrmions emerge, and in remanence non-trivial magnetic textures survive. This demonstrates the possibility to prepare skryrmion-hosting magnetic films on superconducting substrates.