Listed as the Dolphin in 1861 census and 1869, at the same address, and in 1848; by 1895 it is again listed as the Dolphin as it remains for much of its life. So, more reasonably, this is the Dolphin which was called the Wool Exchange tavern in a period about the 1880s?
St Stephen Coleman index
Historical London public houses, Taverns, Inns, Beer Houses and Hotels.
Residents at this address.
1827/Thomas Oldacre, Dolphin, Coleman street/../../Pigots Directory
April 1827/Thomas Oldaker/Dolphin/../../Licensed Victuallers Association
 1839/James Nichols/Dolphin/../../Pigots Directory 
 1841/Robert Cole/Dolphin/../../Post Office Directory 
 1842/Rich Turner/Dolphin/../../Robsons Directory 
 1843/Joseph Baker/Dolphin/../../Post Office Directory 
 1848/Henry Youngman Buckland/Dolphin/../../Post Office Directory 
February 1848/Henry Youngman Buckland/Outgoing Licensee/../../Era 
February 1848/Maria Dunlop/Incoming Licensee/../../Era 
1851/Joshua Henry Wright/../../../Kellys Directory 
 1851/Joseph Henry Wright/Licensed Victualler/37/Edmonton, Middlesex/Census 
1851/Jane Wright/Wife/36/East Barnet, Hertfordshire/Census
1851/Ann Hardwick/Visitor, Dressmaker/34/East Barnet, Hertfordshire/Census
1851/Charlotte Pickton/Servant/17/Southgate, Middlesex/Census
 1856/Joshua Henry Wright/../../../Post Office Directory 
 1861/Joshua Henry Wright/Licensed Victualler/47/Edmonton, Middlesex/Census 
1861/Charlotte Pickton/Housekeeper/27/Southgate, Middlesex/Census
1861/Jane Pickton/Barmaid/17/Southgate, Middlesex/Census
1861/Sarah Withers/Cook/44/Enfield, Middlesex/Census
 1862/J H Wright/../../../Licensed Victuallers Association 
 1869/James Smith/Trustee/../../Licensed Victuallers Association 
 1869/James Smith/../../../Post Office Directory 
 1881/Robert Ludgate/Licensed Victualler/40/London, Middlesex/Census 
1881/Jane D Ludgate/Wife/30/London, Middlesex/Census
1881/Edith Baker/Visitor/25/London, Middlesex/Census
1881/Kate Wetherley/Barmaid/23/Greenwich, Kent/Census
1881/Edith Lane/Cook/20/London, Middlesex/Census
1881/Emma Ball/Housemaid/20/London, Middlesex/Census
 1884/Frederick Whitaker/../../../Post Office Directory 
Dolphin :
 1891/Charles Henry Fry/../../../Post Office Directory 
 1895/Edmund John Coste/../../../Post Office Directory 
 1899/John Remington Martin/../../../Post Office Directory 
1906/Mrs Annie Mary Dacey/../../../Post Office Directory 
1910/Cope & Co/../../../Post Office Directory 
1921/Cope & Co/../../../Post Office Directory