Costco – Southampton – Opening Hours & Store Details You’ll find Costco situated at a prime place on Regents Park Rd, a 1.52 mile drive west from the centre of Southampton (near to Claremont Crescent). The store is proud to provide service to patrons within the districts of Maybush, Freemantle, Upper Shirley, Shirley Warren, Shirley, Redbridge, Wimpson and Millbrook. This page will give you all the information you need on Costco Southampton, including the hours of opening, street address, email contact and other info.