They look mousy in a big way (4)
I believe the answer is:
rats
'they look mousy' is the definition.
The definition suggests an adverb but the answer is not.
'a big way' is the wordplay.
'a big' becomes 'rat' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'way' becomes 's' (Compass direction - South).
'rat'+'s'='RATS'
'in' acts as a link.
Can you help me to learn more?
(Other definitions for rats that I've seen before include "Heels" , "First word of 'The 21 19'" , "Treacherous rodents" , "Damn" , "Rodent pests" .)