You don't really want your horse to "swing" around the barrel with his hind end, you want it to be more of a one fluid motion.
As far as using his back end more, I am not a huge fan of doing roll backs but they work for some.
Try taking it back to ground zero.
Go slower through the pattern and make him stop and put his butt into the barrel where you would want him to "sit" when you are going into your turn.
Do you understand what I am saying?
Do this at a walk, trot, then a canter. If you teach your horse this then he should do this when you ask for more speed. This is engaging his back end and asking him to sit on his hind end and push off from his hind end.
I really hope I explained this correctly, it is so much easier to show then to tell.
But like someone stated above, pics and videos would be great! :)
If you would like I can record a quick video in the next few days to try and explain what I mean better.