Middlesbrough Reference Library holds many thousands of historical sheet maps. Of most value to the family historian are the Ordnance Survey Sheets for the later 19th and early 20th centuries at 6 inches and 25 inches to the mile for those areas of Yorkshire North Riding and Durham which cover Cleveland and Teesside and particularly Middlesbrough. For Middlesbrough only there is an historical set at 126 inches to the mile which are excellent for locating all these little closes and yards mentioned in the Census returns.
 

Holdings are:

North Riding 6 inches to the mile 1856/57; 1895; 1919/20; 1930
North Riding 25 inches to the mile 1894/95; 1914/15; 1928
North Riding 126 inches to the mile Middlesbrough only 1894
Durham 6 inches to the mile 1861/62; 1898; 1921; 1938
Durham 25 inches to the mile 1858/59; 1897; 1919

Remember, apart from these there are many other maps local to the area so please keep asking and keep talking to your librarian.
 

Other Related Resources:

Ordnance Survey Landranger Gazetteer of all 256,000 names shown on Ordnance Survey 1:50,000 scale (one and a quarter inches to the mile) Landranger Maps for England, Wales and Scotland. Ordnance Survey 1994. Microfiche. Filed with the Reference Library’s set of Landranger maps.

Old Maps Co.Uk http://www.old-maps.co.uk
Provides online access to Britain's most extensive digital historical map archive which is jointly owned by Landmark and Ordnance Survey, Britain's national mapping agency. Ordnance Survey has one of the largest collections of historical mapping in Great Britain and until recently this was only held as a paper archive. As the result of a joint venture between Ordnance Survey and Landmark Information Group an extensive digital archive has been created of 19th century 6 inches to the mile maps for the whole country. It is searchable by place and offers a gazetteer as well. As well as the archival map you can view the modern equivalent and the aerial photo.

Oxford Companion to Local and Family History, edited by David Hey. OUP, 1996. pp297-299;333-334. 929.1072041.