When some actions are on the NLA, you can load another action in the action editor using its dropdown menu (or create a new blank one).
Its influence on the rig is determined by the blending mode set in the Edited Action tab of the NLA (Replace is for standard animating, combine can be used for layering animations).
Once finished editing the new animation, you can press the "Push Down" button to clear the action editor and directly have the new animation as a new strip in the NLA.
Selecting a strip in the NLA and pressing TAB, the selected animation is opened in the action editor, and can be modified. Press TAB again in the NLA to close the editing session.
If the blending mode is "Replace" and the extrapolation is "Hold", only the data in the action editor are read by the selected rig, while its NLA strips get muted.