There's a 1940 Bartholomew street map at
http://www.maps-of-london.com/map-hammersmith.htm
The relevant part of it looks like this:
Underneath the letters ITH of Hammersmith, there are 3 unnamed streets that run south from Great Church Lane to the railway, between Elric(?) Street and Colet Gardens. On my 1956 version of this map, they are named as (from west to east) Montgomery Street, Wilson Street, and .... The last one, which I think may be the one you're thinking of (between Wilson Street and Colet Gardens), isn't named!
The area was partly obliterated when Talgarth Road was extended and the Hammersmith Flyover built around 1960.
It should be on one of the Alan Godfrey maps, sheet 86 I think, unfortunately not in my collection. Anyone got that sheet?