A previous paper by the authors reported on the estimation of resonant heave (springing) damping of tension leg platforms (TLPs) and a method of scaling for full-scale prototypes. The present paper is a sequel to this work, and examines the effect of adding an appendage in the form of a disk to TLP columns, and the influence of a small uniform current. Model tests conducted on a cylinder + disk in heave show that the heave damping induced by the disk is linear with the amplitude of oscillation. The disk is found to increase the form drag coefficient twofold, in accordance with published results based on isolated edge theory. The effects of a small uniform current were also examined during the model tests. Results show an increase in heave damping ratio that is linear with the current velocity. In the presence of a disk, the damping induced by the current is doubled as well. Scaling laws are proposed in this paper that enable extrapolation of heave damping due to appendages and small currents to full scale. An example calculation shows that for a full-scale TLP column, the heave damping is increased by about 300 percent due to addition of the disk, and by 87 percent due to a small current. The combination of the disk and the current increases the heave damping of the column by a factor of six.