The first thing I do is move each yarn tail in the direction it came from (on either side of the row join).
Now I start doing reverse duplicate stitch on the back of the work. I always make sure to pull the yarn tail in the OPPOSITE direction it was coming from to help match up the joins on the right side of the work and get rid of any holes.
Reverse duplicate stitch on the back of stockinette is two diagonal actions with a tapestry needle threaded with the yarn end. Each diagonal action catches two purl bumps and as you go along the row you go through each purl bump twice which makes it virtually impossible for yarn ends to work their way loose.
Here is the second action. It is hard to show this properly because the various yarn floats at the back of the work get in the way. I just weave around and over them as I can but I suspect you could make this process completely invisible if you meticulously did reverse duplicate stitch UNDER all the floats.
You really don't have to do this for very many stitches as it does an excellent job of securing the yarn ends. I generally cut the yarn tails pretty close to the work as I'm confident they won't work their way loose.
Voila! You're done. There are some sections where you can see the yarn tail diagonally crosses some yarn floats but the actual reverse duplicate stitch isn't noticeable. Even better, it isn't noticeable from the right side of the work.
Theresa has done a great job of using both a photograph and some computer magic to show reverse duplicate stitch on stockinette HERE.