One of my first vintage bicycles was a 1927-E Westfield built Shapleigh motorbike. Like your bike it came without a head badge. I used online pictures of others' bicycles, and online pictures of various period catalogs to determine both the manufacturer and retailer, (as well as the date code). Often the old catalog pages are offered for sale online; although I prefer to collect the real article, once I have purchased a set of Shapleigh catalog pages, (perhaps because they were in color). Sometimes I download catalog pages from this site and others, and I am grateful to those who have shared. For example, the picture you posted, looks like a page-1711 from an advertised 1935 Shapleigh catalog. Below is one of the pictures previously shared by others; 1933-34 Sears, (see bottom of page).
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Ladies bikes were often
blue. Below is an example the Sears Elgin Chicago head badge that I was thinking might fit.
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Be sure to measure the hole spacing and shop-around for badges. Not sure when Shapleigh may have adopted badges with the vertical holes.