Quine's thesis about the inscrutability of reference has nothing to do with context-dependence. As you noticed yourself, the context is just one more means to fix a reference. It is not a reason for skepticism about the nature of reference.
Rather, Quine's inscrutability of reference thesis comes from one aspect of the gap, in inferences from the particular to the universal. Faced with a particular object, how do I find the right general terms to describe it? Is this a "rabbit", or maybe "undetached rabbit parts", as in Quine's famous "gavagai" example?
An inference from the particular to the universal has been called induction. And indeed, Quine's inscrutability of reference thesis is akin to Nelson Goodman's New Riddle of Induction. Put briefly, the "old" riddle of induction was: faced with some particular green (e.g.) objects, can I justify a thesis about green objects in general? The "new" riddle of induction is: faced with some green objects, is it correct to describe them as 'green', or rather as (e.g.) 'grue', for purposes of generalization? When grue = green until date XYZ, blue hence. Quine argued that many possible conceptualizations are equally correct. That there was no fact of the matter which conceptualization was more correct. Goodman tried to explain, how we in fact choose one candidate conceptualization over another: the logic of induction.