It would seem that what is more often called ‘Dead Man’s Well Path’ was referred to as ‘Dead Well Path’ in the Survey Of Public Rights Of Way, 1949:
Dead Well Path, Warminster, is mentioned in the Survey Of Public Rights Of Way under the National Parks and Accesss to the Countryside Act 1949, 1950-1959 and an Unofficial Commencement on a Public Footpath Map under the Rights Of Way Act 1932, 1935. G16/132/51