In this case, I did not think it was necessary to tie any sort of knot to connect the new thread to the old. I simply started a new ring, and put the old end through in the same direction as the core of the new ring. The photo above shows the first half of the first double stitch.
In the photo above, you can see the old end sticking out of the ring. It's best to just leave it while you tat some more past that area, before trimming the end.
The new end that was left by adding the new shuttle thread, is now being worked under the chain.
Here you can see both of the thread ends sticking out of the new ring and chain. When more tatting has been done past that area, give the old ends a firm tug before trimming them off, to make sure the all the tatting is firmly in place.