1. It seems that the subject of dealing with the tail end of the thread is one that arises frequently. Tatters all strive to find ways that are simple to do but effectively hide the tail. Jane Eborall has prepared some new diagrams which will help the new tatter learn these techniques.
If you are tatting a pattern where the last ring joins back to the first ring made, then plan ahead by preparing the initial ring to hide the ends by means of helping loops.
These loops may be a thread that is a size smaller than that being tatted or, use a mono filament thread, such as invisible sewing thread or fishing line, or a dental floss threader. Use two pieces each about 6" in length. Fold in half and knot ends together. (I also place a safety pin into the knot.) The loops will be inserted into the tatting so that the bend of one loop hangs down on each side of the ring with the ends toward the center of the ring. Over tat the helping loops. At about the midpoint of the ring add in the second helping loop.
Close ring very carefully and secure the hanging loops with tape, safety pins or paper clips so that they will not come out of place while you tat the length required. When finished attach the last chain to base of initial ring. Cut leaving 6" tails. Insert one tail into each hanging loop and gently pull the loop and tail into the ring. Completely pull thru and trim closely allowing the end to recede into the tatting. Be very careful when cutting.
2. Hold both tail and shuttle thread together taut and tighten the half stitch.
Note that the tails enter the loop following the shuttle thread alternating from one side to the other. Also remember to tighten the stitches extra tight as you are covering more than one thread. Do this for the complete length of the chain. This method is almost invisible if you keep good tension when tightening the stitches.
3. Thread ends may be "woven" or "sewn" back into the double stitches. Place thread tails on fine blunt tapestry needle and weave under the "bars" of the double stitches around the ring or chain. Trim close and allow end to recede back into the tatting. Again, take each tail in opposite directions.
Adding in the new thread while still tatting the last ring is one way to deal with the tail end while the tatting is still in progress. (Matthew Takeda's method which he shared online.)