A nice way to create secondary action and animation in Maya is to use dynamics. This is an alternative to manually keying the secondary action yourself. Sometimes it can look much more natural. Sometimes it can come as “free” animation. Other times, it comes at a cost. It can be difficult to set up. Especially on character rigs that you might not be familiar with.
There are some tools that can help. a new one from Jeremy Ernst is designed to make overlapping and secondary action much simpler to create dynamically. Called the Dynamic Motion Motivator, the script allows you to select objects or controllers and make them dynamic.
The script lets you create more than one Dynamic motion Motivator (DMM) system, and edit the aspects of each system. Once you get the look you want, the script lets you bake down the dynamics into keys.
If you are looking to simplify secondary action or overlap, check out Dynamic Motion Motivator which can be purchased for $30.00.