An aside, ThumbelinaPM - although they didn't start till the 1870s, do you have the First Series search sheet for the [15s] Lands [of old extent] of Easter Glentore? Do you know which estate they formed part of? That estate's records may still have the original dispositions/feu charters/leases on which these sasines are based.
The NRS have the following records, sadly no maps.
BR/NBR/8/1087 General Manager's file re siding for Easter Glentore Colliery 1865 This record is Open.
CS97/93/B4A Easter Glentore, East Markland, Lanarkshire 1820 This record is Open.
CS271/43542 Bill of Suspension and Interdict Arthur Wilson (Willson, Wilsone), writer in Edinburgh, heritable creditor to John Arthur, portioner of Easter Glentore v John Fleming of Craigs, & another (Trustees of John Arthur of Easter Glentore) 12 May 1788 This record is Open.
RH9/4/121 Inventory of James Whyte's writs, 1667-1726, of lands of Wester Glentore in parish of Easter Monkland, sheriffdom of Lanark 1731 This record is Open.
RH9/17/362 Tacks (leases) and missives of let: University of Glasgow to Nicol Spence, writer in Edinburgh, of parsonage teinds of his half of the easter 16s. land of Glentore in parish of New Monkland, sheriffdom of Lanark, for term of 13 years 15-16 Feb 1738 This record is Open.