I'm using Bottom Line by Superior Threads - You can buy bobbin assortments of the different colors, but I mostly use a light and medium grey and a light and medium taupe - those four colors almost disappear in nearly all the fabrics I'm using in one quilt top.
The other quilt top I'm doing is a little more exacting, and I'm using a rose pink, a brick red, a leaf green and a hunter green for those pieces. They're invisible.
I had tried Masterpiece, Mettler, YLI 100 silk, clear nylon and smoke nylon - if they weren't invisible, they shredded or snarled easily or kept falling out of the needle. The Bottom Line threads handle a lot easier.
If you're already using a very fine matching thread, maybe you're just not bringing your needle up close enough to the fold? I don't know how experienced you are, so I just mention that as a possibility. I wasn't doing that at first, either and the stitches were showing more than I wanted them to. I think it just takes practice.
I asked the ladies at the quilt store if they thought I was doing okay with the stitch size and the owner laughed and said, "Okay!? They're microscopic, are you kidding!?" That made me happy because, not knowing what I'm doing, I was afraid I was stitching too big. So maybe you just need to catch a couple fewer threads when you come back up to the top?
Good luck - hope it's lots of fun for you! :)