The same three-letter word can be inserted into each of the following words to create new ones:

BREAD
WIRED
BAD
FEAR
CLOD
WEAR

HINTS:
#1: With the exception of WIRED, the three-lettter word is inserted before the last letter of the clue words.
#2: The three-letter word begins with T.

Answer:

The missing three-letter word is THE:

BREATHED
WITHERED
BATHED
FEATHER
CLOTHED
WEATHER 

Credit: LA Times Crosword
Sylvia Bursztyn and Barry Tunick
PARDON THE INTRUSION