I don't know if you can really see the textures in the photos below, but it is, I think, similar to what you're describing. Both photos were taken by Bill Keunzel of the Detroit News. They're on a more thicker, textured paper than typical newspaper photos, especially of the time. I think it was used when the photograph was to be displayed or framed. I have a few others that are on similar paper. I think it was also used in portrait studios for baby or family photos. If you have a family photo done by a professional photographer from the 1930's or even much later, it might be on similar paper. I am NOT an expert, however, so no guarantees!
First one is the Detroit Yacht Club from around 1930. Second is U-M vs. Harvard 1929 at U of M.
detroit yacht club_hq.jpg
um photo2 copy_hq.jpg