I might catch hell for this... but...
I always lunge in side reins. Every time.
Before anyone flogs me, let me explain why...
The purpose of lunging is not to have your horse run in circles. It is to get them to a)get some energy out and get the horse to focus before the rider gets on, or b) work the horse.
The best way to focus a horse is to get his mind on working, and make him use himself correctly.
The side reins act as something for the horse to push into, giving them some support (not something to lay on). Make sure you have sufficient push from behind with your whip. In return, it encourages the horse to use their back correctly and carry themselves in such a way that the correct muscles are developed.
A horse without side reins, particularly in the situation you are describing, is most likely leaning to the inside, has an outside bend, their head is high, and is rushing. In NO WAY is this beneficial to the horse, nor is it helping to develope any kind of good habits.
With or without side reins (whatever you choose to do), some good exercises to do on the lunge line...
Walk/trot/walk transitions every circle, then every half circle if you really need to get your horse to focus.
Spiral in and out, making sure not to spiral in too tight, and not to hold the smaller circle for more than a circle. (When the horse speeds/slows its tempo it means the circle is as small as it should go and it's time to spiral back out).
Once you've established a good push, rhythm, and focus, go to trot/canter/trot transitions. If the horse gets sloppy, you can handle it one of two ways:
Push the horse forward on an oversized circle and see if they settle and work through it.
Bring him back to trot and regroup, establish a relaxed even tempo, and ask for the canter again.
My horse is 14, and still gets silly at times at the canter. He does exactly what you were describing, so for us what works is the trot/canter/trot transitions until he settles. Plus the side reins.
Good luck :)
"The white horse moved like a dancer, which is not surprising: a horse is a beautiful animal, but it is perhaps most remarkable because it moves as if it always hears music."