The evolutionary path involved the car's motor, radiator, and so forth being moved forward, the engine overlapping the front axle-line whereas before it rode entirely aft of that point. Headlamps could not continue to be placed forward of the grille, so they remained about where they were (in relation to the axle line), but now were beside the radiator rather than in front of it. Since visual (and some actual) streamlining became the styling fashion of the 1930s, headlight housings were usually shaped like bullets, artillery shells or teardrops -- pick your favorite analogy.
These assemblies were mounted various ways. Some were attached to the radiator/grille housing just forward of the hood panels. Others appeared to be sitting atop the front fenders. Often they were nested between the hood and fenders, atop stanchions that were mostly hidden by front end sheet metal.
Most manufacturers made do with a single headlight assembly design for each car brand, or even shared them across brands in the interest of economy. For some reason Studebaker, which was hit harder than many firms during the Great Depression, opted for two headlight housing designs for its 1938 lines. This link mentions that President and State Commander models shared capped headlights featuring more angular shapes, whereas regular Commanders carried the more common bullet shapes.