In this paper, we compute the cohomology class of certain 'special maximal-rank loci' originally defined by Aprodu and Farkas. By showing that such classes are non-zero, we are able to verify the non-emptiness portion of the Strong Maximal Rank Conjecture in a wide range of cases. As an application, we obtain new evidence for the existence portion of a well-known conjecture due to Bertram, Feinberg and independently Mukai in higher rank Brill-Noether theory. Contents