Hard/soft magnetic bilayer thin films have been widely used in data storage technologies and permanent magnet applications. The magnetic configuration and response to temperatures and magnetic fields in these heterostructures are considered to be highly dependent on the interfacial coupling. However, the intrinsic properties of each of the layers, such as the saturation magnetization and layer thickness, also strongly influence the magnetic configuration. Changing these parameters provides an effective method to tailor magnetic properties in composite magnets. Here, we use polarized neutron reflectometry (PNR) to experimentally probe the interfacial magnetic configurations in the hard/soft bilayer thin films: L10-FePt/A1-FePt, [Co/Pd]/CoPd, [Co/Pt]/FeNi, and L10-FePt/Fe, all of which have a perpendicular magnetic anisotropy in the hard layer. These films were designed with different soft and hard layer thicknesses (tsoft and thard) and saturation magnetization (Mssoft and Mshard). The influences of an in-plane magnetic field (Hip) and temperature (T) are also studied using a L10-FePt/A1-FePt bilayer sample. Comparing the PNR results to the micromagnetic simulations reveals that the interfacial magnetic configuration is highly dependent on tsoft, Mssoft, and the external factors (Hip and T) and has a relatively weak dependence on thard and Mshard. Key among these results, for thin tsoft, the hard and soft layers are rigidly coupled in the out-of-plane direction and then undergo a transition to relax in-plane. This transition can be delayed to larger tsoft by decreasing Mssoft. Understanding the influence of these parameters on the magnetic configuration is critical to designing functional composite magnets for applications.