Can I further ferment store-bought kim-chi?
First, as noted in Sneffel's answer, "real" kimchi is eaten at a variety of fermentation levels.

It is possible the fermentation would resume in store-bought kimchi, but only if it is raw and unpasteurized.  A lot of store-bought kimchi is pasteurized (depending on where you buy it), which would kill off the bacteria that would typically be used for fermentation.  Kimchi that is not pasteurized would typically be sold in a refrigerated section and would perhaps have a container/lid that would allow for gases to be released as fermentation continues.  (The only real way to tell would be labeling or contacting a manufacturer.)

In general, I would not advise trying to resume fermentation on a store-bought product, as you don't really know the conditions it was made under or whether continued fermentation would proceed safely.  (Chances are that commercial kimchi would be processed in a way to ensure its safety, even if stored at low but not refrigerated temperatures, but again, this is difficult to know.)  

Kimchi isn't that hard to make to begin with, so if you can't find what you like in the store, I'd suggest just making it yourself and letting it ferment to the stage you prefer.