"America's most popular luxury brand in the early 1930s was Packard. By 1933 it was suffering from reduced sales so much that for model year 1935 it launched a model for the upper-middle price range, the One-Twenty. It was a sales success, probably saving the company, which then continued on into the mid-1950s."
The 1935 inline eight cylinder 120 was Packard's entry-level line. For 1937, the 6-cylinder Packard Six was introduced as the entry point. Model year 1938 saw new bodies for those models. The Packard Six retained its name, while the 120 became the Packard Eight. (But the Eight became the One-Twenty again for 1939, and the Six was renamed Packard One-Ten for 1940.)
The Wikipedia entry for the One-Twenty is here.
By the late 1930s there were comparatively few remaining upscale American automobile brands. The early 1930s saw the disappearance of Marmon, Franklin, and Stutz. By 1938 Auburn, Cord, Duesenberg, and Reo were gone. Other carmakers had rethought market placement of their models. So the number of 1938 competitors for the Packard Eight is less than for the 1935 One-Twenty that I featured in the post linked above.
Unless noted, images below are of cars listed for sale on the Internet.
1938 Packard Eight Touring Sedan
What we see here is mainline mid-late 1930s American sedan styling abaft of the cowling/A-pillar. The hood/grille design is standard 1930s Packard, while the front fenders are generic mid-late 1930s. The hood ornament is inappropriate for this restored car.
1938 LaSalle 50
Facelifted 1937 design. Little character difference from the Eight on the after part of this body, though windows are larger and less-rounded.
1938 Buick Century Touring Sedan - Gallery Aaldering photo
The same can be said for this other General Motors model.
1938 Lincoln Zephyr
Zephyrs, launch for 1936, featured stronger "streamlining" shaping and details. Front ends were facelifted for 1938.
1938 Chrysler Imperial Touring Sedan - Barrett-Jackson photo
Facelift of a 1937 design.
1938 Packard Eight Touring Sedan
Side-view comparison to the Buick Century.
1938 Buick Century Touring Sedan - Gallery Aaldering
By the late 1930s, GM was clearly the American styling leader, so the Packard Eight and Buick Century 4-door sedans resemble one another fairly closely.
1938 Packard Eight Touring Sedan
Note the two-pane backlight window and the shape of the trunk compared to the Buick below.
1938 Buick Century Touring Sedan - Gallery Aaldering
Also a two-pane backlight. The trunk is larger, but in the same spirit. Door hinge positions are virtually the same on both cars.
To summarize, Packard's 1938 redesign was in line with that of its strongest competition (Buick and LaSalle). However, those brands received airier, crispier, passenger compartment greenhouse styling for 1939, making the Eight's/120's more rounded tops and windows a little old-fashioned.