In part 4 of this saga, after having followed the first two legs of the grief map (denial and anger) as outlined by Kübler-Ross, I took a side trip to magical thinking which allowed me to bypass bargaining and depression. Eventually, however, rational thought once again reared itself, and I decided on a course of action that allowed me to achieve a state of measured acceptance, and move on to what is now Part 5 of this project.
I came to realize that all of the work I had done had to be undone, and that undoing all of my work was the most reasonable way to salvage the project. I also came to realize that while the balls of yarn looked pretty all piled together, I both wanted and needed a rug. I did not particularly want or need myriad balls of yarn to display.
And while I did manage to find a square that I both wanted to make and was sturdy enough to be used in a rug, the new squares required more yarn per square than the squares I had unraveled. Consequently, it took yarn from just over three of the previous squares to make one of the new squares.
Given that the new stitches were made with a much smaller hook and the stitches were very tight, I thought that weaving in additional ends would result in lumpy, non-uniforms squares, and since I planned to be able to walk on the rug, I did not want lumpy squarea.
I looked into my trusty crochet tool kit of stuff:

and without too much trouble, found my trusty felting needles.
With the aid of a felting needle, a piece of foam, and a sharp pair of scissors, I was able (in just three protracted steps) to rejoin the ends of the yarn so that I could crochet with it and not have the added weight of the ends.
Here is the process I used:
Take two strands of matching yarn:

Split each strand in half for about 3 inches:

and trim one of the splits from each strand of yarn. Roll the two remaining, thinner strands together and then pierce with a felting needle until a firm join is formed:

The finished product:

So I, like the ill-fated Donner Party, learned the hard way that short cuts have their own perils.
This is an awesome method!
I have to keep that in mind 🙂
It does work really well. 🙂