She left her role last month, according to LinkedIn.
Various reports during the summer indicated that more than 1,300 jobs would go across the business, including more than 300 head office roles.
Cool, who was appointed in May last year, was tasked with overseeing media relations, public affairs, sustainability, community and the Asda Foundation.
This work had previously been handled by Freuds until the retailer decided to serve notice on its contract with the agency in September.
Following its split with Freuds, Asda appointed PR agency Mischief to a food-and-drink brief in January. Asda has been contacted for comment.
Before joining Asda in May last year, Cool was group head of corporate affairs at John Lewis, where she served for almost 10 years.
Prior to this she was executive director at the Commercial Broadcasters Association between 2003 and 2006. Cool was also a researcher and press officer for the Labour Party between 1994 and 1997.
Cool declined to comment.