The back side won't show anyway and when you work the ends in it won't look so bad.
That said there are some tricks. If you are only carrying a yarn a short ways you can carry it up the side of the work catching the color you are carrying up under the color you are working with each time you are at the side where it is waiting. You can carry yarns a long ways this way but I don't like to much, especially if I end up needing to carry several colors along a side.
Another thing you can do is to introduce new colors the way you have been but knit over the tails as you introduce them. Have the tail you want to work over up next to the holding needle and insert your working needle so that it knits the stitch you are on but goes under of over the yarn tail. Alternate knitting under and over the tail for a couple of inches, then pull on it to even out the tension and stretch everything a little and then you can cut the covered tail off. This works pretty well, but if your colors are highly contrasting the hidden color may show through a little. Then I prefer just to cut off the tails and work them into the matching color stripe.
I used to hate working in tails but I kind of like it now. Not a lot of thinking involved and you can relax with your tail work. LOL I did a little girl's dress once that I had at least 200 tails to work in when done.