Call them even more insensitive for revealing how person can be contacted (9,6)
I believe the answer is:
telephone number
'even more insensitive for revealing how person can be contacted' is the definition.
I can't tell whether this definition defines the answer.
'call them' is the wordplay.
'call' becomes 'telephone' (synonyms).
'them' becomes 'number' (I can't justify this - if you can you should believe this answer much more).
'telephone'+'number'='TELEPHONE NUMBER'
Can you help me to learn more?
(Other definitions for telephone number that I've seen before include "that may be kept in cell" , "Very large figure" .)