There might be a way to do this by actually working flat -- i.e., in rows -- but wrapping the first and last stitches of each row to maintain the round construction. It's a method used by colorwork knitters who don't want to carry an unused thread all the way around a sock or garment knitted in the round. However, it will leave a seam or line on one side of the sock. And while this seam is nearly invisible to the naked eye, it may not FEEL invisible to the naked foot!
You can find more detailed directions on the internet, but here's how it works:
- Work to the end of your round.
- Wrap the first stitch of the next round and return it to the left needle.
- Turn your work so you are working the wrongside, back over the stitches you just worked.
- Work the wrong side all the way around to the stitch you just wrapped and work it.
- Wrap the next stitch (first stitch of the previous row) and return it to the left needle.
- Turn your work so you are working the rightside again.
- Work the right side all the way around to the stitch you just wrapped and work it
- Wrap the next stitch and return it to the left needle.
- Turn your work.
Continue working rightside/wrongside and wrapping the stitches as described until you have completed the section in question, then return to working the sock in the round. Using this method, you could work your reinforcement into just the sole of your sock by picking up and dropping the reinforcement yarn when you reach each side of the foot.
I'm not sure my instructions are very clear, but it should give you a rough idea of what I'm trying convey. Good Luck!