I will like to know whether we are doing this correct.

What happens when a player wants to use the non-dominant side of an Action Card in non-combat, non-social encounter mode (e.g. during an open-ended chapter, where the party searches a series of rooms in a dungeon)?

We play it like this: Characters are always in neutral stance outside combat or social encounters. Therefore, a player is allowed to change her stance before using the non-dominant side of an Action Card, but this is not mentioned in the rules. Alternatively, the player is not allowed to use the non-dominant side outside combat or social encounters.

Have we overlooked something? What do you do???

It's up to the GM if he wants to let the players use a stance in story mode (or starting in encounter mode). I would let a player use a stance if he had a good and valid RP reason for it. But yes, unless the GM allows it, it is impossible to use the non-dominant side in story mode.

I agree with gruntl's assessment of the situation. It would be up to the GM to determine if using the non-dominant side is possible.

Personally I would require a compelling reason or situation to justify the change from the dominant stance.