There was a Six Pit at Lumley Colliery and also a Sixth Pit at Beamish. (Source: Durham Mining Museum website.)
I found several other examples, spread across he country. If you think about it, by the end of the 19th century mining companies had grown large enough to own several mines, and would sink new shafts even in the larger mines. These would initially be numbered, though often they had names as well. Most mines had fewer than six shafts, but there were enough bigger ones that it was hardly rare. Whether the locals would say "six pit" or a variation on that is hard to say, as it might not match official usage.
What is unusual in the picture, and would allow identification:
1. The engine house with its hipped roof, and that dormer thing where the cables pass through.
2. The pithead tower, with no superstructure over the wheels, and vertical uprights seen from this angle (though I think angled seen from the left side).
Of course you have to find a picture of this one before you can recognise it!
