The Definitive Dwiggins no. 165 addendum—Another B
On page 114 of W.A. Dwiggins: A Life in Design by Bruce Kennett (San Francisco: Letterform Archive, 2018) the author shows a cropped image of an insert for Beckett Paper from Direct Advertising vol. IV, no. 3 entitled “The Buckeye ‘Dummy’ Covers”.  “Dwiggins cleverly recycled the ornamental B from his Bartlett days, Kennett comments. Unfortunately, the decorative B, despite looking very much like the one Dwiggins designed for Alfred Bartlett in 1906, is not by Dwiggins.
This B differs in a number of subtle ways. The most obvious is the addition of a swash at lower left. A cadel stroke is missing from the large loop at the bottom right; and various strokes are slightly different in curvature (e.g. the loop inside the large loop at the upper left) or sharpness.
The Buckeye B is not simply a revision by Dwiggins of earlier artwork but a redrawing by someone else using his Bartlett dodger as a template. I believe that it is not by Dwiggins because 1. the textura of the title is not his style, 2. he eschewed swashes in his calligraphy, and, most importantly, 3. he never recorded having done any work for The Beckett Paper Company in his account books.
I believe the Beckett insert was designed and lettered by Dwiggins’ art school classmate Oswald Cooper. The textura looks like his work (see the stationery he designed for Ernest P. Thurn in 1914 or earlier). Furthermore, Cooper had been doing work for The Beckett Paper Company since at least 1915 and was still doing work as late as 1920. 
1. The Beckett Paper Company of Hamilton, Ohio offered Buckeye Dummy Covers to help printers by “suggesting appropriate color combinations, facilitating the making of attractive dummies and minimizing the necessity for experimental press-work…”
2. The Oswald Bruce Cooper Papers (Wing Modern MS Cooper) at the Newberry Library contain artwork for several jobs for The Beckett Paper Company, including for “A Simple Job” (1915) and greeting cards for 1919 and 1920. See Series 2 Design Work.