I don't think this takes us much further, but here is what I have found:
There is a newspaper item from 1919 (Motherwell Times, 14th November) referring to the silver wedding of Mr and Mrs Quigley of Kilmartin's Land, Carfin. A later item, from 1940 (Motherwell Times, 12th January), reports the death of Annie McKendrick aged 64–the widow of the late Patrick Quigley–of Kilmartin's Land, Carfin.
In the 1915 valuation rolls I find Patrick Quigley, miner, as the occupant of a house at Carfin. The proprietor is William Kilmartin, spirit merchant, Woodside House, Hamilton.
The order of the records in 1915 is:
Byres Knowe [Carfin Byres, just to the south?]
then a block of 13 occupancies, mostly miners, (prop. William Kilmartin) starting with a spirit shop occupied by William Blair
then a block of 4 occupancies including Patrick Quigley (prop. William Kilmartin) starting with a spirit shop which looks to be occupied by William Kilmartin
then Union Place
In 1921 the Quigley family is indeed found at Kilmartin's Land [there are 16 households in the census with this address]
The entries for Kilmartin's Land are preceded by Back Row and followed by Mason's Land [the 1915 valuation has proprietor John Mason, grocer and miner, occupying a house and shop in the block of 13)
When Patrick Quigley died in 1934 his home address was listed as 67 New Stevenston Road, even though his widow's address in 1940 is Kilmartin's Land. This supports the suggestion that Hamilton's Land was on New Stevenston Road.