Some years ago and perhaps still, there was a Gifford family that lived in a beautiful house just north of the cut (eastern most) in the woods east of McPherson's Ridge at the end of a long lane that begins where Doubleday Ave and Wadsworth Ave intersect (running south); first day's battle field.
I've never heard of a woodlot referred to as Gifford's Woods during or after the battle. Perhaps in more contemporary times someone referred to the heavy woodlot there as 'Gifford's' as a local reference point only, ie....those that know the area.