On the east side and parallel to New Bridge Street lies Waithman Street. Not a busy thoroughfare, just a bit of pavement leading from the top of Pilgrim Street to Black Friars Lane. You can get to it from New Bridge Street, but that means climbing the stairs; it is easier to turn into Pageantmaster Court from Ludgate Hill, then right into Pilgrim Street and then left as you hit the bollards at the top. Google Street View is no help as the camera car apparently did not like to attempt the narrow street sections in the block, but here is the map to help you. Waithman Street is situated between the two blue markers.
And this is what you’ll see if you have managed to get to the end of Pilgrim Street.
The back of 100 New Bridge Street shows large tiled panels, designed and hand-made in 1992 by the artist Rupert Spira (1960-), the only tile commission of his in England; he did another one in a garden in Paris (in 1991), but went back after this short tile experiment to designing and making ceramics.(1) See for more information on his work his website here. The designs of the wall tiles remind me of the work by the Dutch artist M.C. Escher (official website here).
Waithman Street was named after Robert Waithman (1764-1833), linen-draper and councilman for the Farringdon Without Ward. I have not found out who proposed the naming of the street, but it was motioned in a Building Act Committee Report of a meeting on 30 July 1891 as item 66 “that the thoroughfare at the back of Ludgate-hill station between Pilgrim-street and Union-street be named Waithman-street”. It would seem from the wording that this stretch of road had no previous name. London County Council approved the motion on 29 September 1891(2) and the approval was subsequently registered for 1891 in the List of Streets and Places within the administrative County of London .
(1) Information from https://tilesoc.org.uk/tile-gazetteer/city-of-london.html
(2) London County Council, Minutes of Proceedings, p. 917 (Guildhall Library)