By grabbing a ring from above, Simon can suspend himself above the ring. Letting go in that position will give him a huge vertical boost. Oddly enough, this boost is higher when his sprite points to the right. This can be used for a couple of shortcuts in various levels. Holding Left or Right will make Simon enter the regular swinging motion.

If you can't jump high enough to grab a ring like that, this is what you can do -

*Jump as high as you can towards the ring.