I try to avoid, but for a double irish chain, I know the diagonals call for quilting through them. So, just take it one stitch at a time instead of stacking several stitches on the needle. It'll take some practice and some work, and they'll get closer, but they won't be quite as close as other areas.
I try to pay attention to the pressing, too. Trying to make sure I don't end up 8 layers of fabric on top of each other.
I think I've heard of people going back over that area and filling in the "missing" stitches (the gaps, basically)???? I don't do this. I just quilt and no one notices....