This is the Linbrig trig point (390m) on Green Side in Alwinton. Although the area is called Linbriggs, this trig point is simply called Linbrig.
From the layby head east back towards Alwinton then head up the bridleway. You can follow the quad bike tracks to the top of the hill.
Just under 2km to the trig point, but over 180m of climbing.
Follow this map, or find it on OS Maps.
Thanks to Ben Gibbinson for the photos and information.