I also worked on softening the edges of the fabric so it didn't seem as harsh. I attached a ramp to the opacity channel under 'Geometry' to achieve this. Out of the three images below, I like #1 the most because it is clearly softer but not too obvious.
I thought I used the texture with the softened edge for the animation above, but it doesn't seem like it had much of an effect. I wonder if I rendered the wrong one by mistake or by decreasing the samples I turned something off accidentally. I plan on investigating and doing more tests both in regards to softening the edges and trying to keep the initial ripple animation visible.