Rex overwhelmed by dad becomes more distant (7)
I believe the answer is:
farther
'more distant' is the definition.
(I've seen this before)
'rex overwhelmed by dad' is the wordplay.
'rex' becomes 'r' (common abbreviation e.g. GR=George Rex, King Georg).
'overwhelmed by' indicates putting letters inside (I've seen this in other clues).
'dad' becomes 'father' (I've seen this before).
'r' put inside 'father' is 'FARTHER'.
'becomes' is the link.
(Other definitions for farther that I've seen before include "Beyond; her raft (anag)" , "Over a greater distance" , "longer" , "More distant" .)