A bit late, but let me ask, why is there a thin boarder polygon? is this intentional (ie different material or outlining), an artifact of how it was first made (stitched together from different parts using cloth simulation), or when you say you are cutting the corners, why are you cutting them, is it to control how the edge looks (ie when rendered does it keep the edge smooth as part of how normals are automatically calculated? or are subdividing for simulation and it breaks due to the corner?)?
Topology 101: use only quads, with permission to have 3 poles and 5 pole corners time to time. you may need to migrate poles around to avoid distortions to the normals. Here is how I would typically do this, especially for cloth and character models.
See the funky square with the 3 in it, that's the only thing that looks ugly to the human eye, but when sub dividing or rendering it smooth, or simulating cloth, this is much better. The main idea is with topology, square are best, square are simple, we can divide squares into smaller squares, avoid triangles unless you really know what you are doing since this breaks most things (such as rendering and simulations) (ie only really use it on the final topology taking advantage of 4 points not being planar to have direct control over final triangulation and surface normals)
The trick is that we want to create grids and loops of squares. loop make the green path. Here I widened the yellow boundry line you had to be the RED, its often smart to try to keep neighboring squares with in a power of 2 less than or greater (ie half size or double size) at most most areas have 4 lines connected to a pole, this is a 4 pole (se the green circle with the 4). these are best/ideal, and when rendered smooth look smooth. to get square loops to curve and merge around each other (Ie the 3 green loops arrows, which there are actually a total of 5 loops made with this topology) you'll need to take advantage of 3 poles and 5 poles. these do have some graphical oddities when rendered smooth, but so long as they are near flat, they stand out the least compared to the corners of triangles, and you want to avoid having a 3 pole connect to a 5 pole. Now depending on how I want to use the topology, I'd make the #3 corner less pointing out, and Id actually pull it in aggressively to allow a smooth curve, similar to what I am doing with the blue for the green loops I am showing.
