So, at this point we have a heel flap and a gusset and a little problem:

The sock has no top. It’s time to put those stitches from the scrap yarn back onto their now empty pegs.
Then there is just closing up the gap along the top of the gusset. To do that I knit just the top/instep pegs, picking up a gusset stitch as I went along.
To do the instep, start a new working yarn. Go ahead and cut the old one, leaving enough tail for weaving in later. Put the instep stitches back on the loom. To set up each row, pick up the closest loop (or pair of loops if you slipped your gusset edges) from each gusset edge stitch and place them on the peg with the first and last instep pegs (Normally that would be pegs 25 and 48, but for my sock it was 24 and 47). For this first row, you’ll notice that there is a pretty big gap between the edge of the instep and the first gusset edge stitch. For at least the first row I picked up loops from the gap instead to minimize holes. Actually, I must have done this for at least three rows before I began to pick up gusset stitches. That was definitely excessive and I think it showed a bit. I think either two rows or maybe picking up two separate loops on the first instep row will be enough. If anyone experiments with that let me know.

Here the top stitches are back on the loom and you can see the gusset stitches that need to be joined. I think the first picked up loop is already on peg 47.
With the new yarn, knit across just those 24 loops that were just replaced, knitting the picked up loops along with the original loops. This seemed VERY bulky on the loom—knitting 3 over 1 on the first and last stitches. I thought it might make an annoyingly stiff or uncomfortable join, but it really wasn’t too bad. I do wonder how it would look if you knit the bottom loop over the picked up ones before knitting the row.
After the first row, pick up another loop (the next closest one to the instep) from each side of the gusset edge and put it on the outermost top/instep peg. Knit back across those pegs. Just keep doing that until you run out of gusset, and you can start knitting in the round again.

You can see the finished closed up gusset after a few more rows.
Next time I think I’ll try a very plain sock so I can really see any holes and joins. I think I’ll also try not slipping the gusset edges so I can just pick up one edge loop. I am afraid that I’ll have trouble seeing which loop to grab, though. I usually do. It’s so much easier for me to see a slipped edge.
Thanks for taking a look at my experiment. This is clearly more complicated than doing a heel flap on needles, but it is certainly do-able and the socks do fit me a bit better than short row heels. So that’s a functional success, but still a design in progress.
