Close-Knit Words

Python 3.5, 68 bytes

w=input()
a=0
while 1:s='%c'*3%(a,a+1,a+2);a+=1;{*s}-{*w}or print(s)

Prints output strings, and terminates with error when the character value gets too large.

Generates all strings of three consecutive and prints those that are a subset of the input word.

Pyth - 11 10 8 7 bytes

Super brute force method.

@^z3.:G

Test Suite.

@            Setwise intersection, finds common strings between the two lists
 ^           Cartesian product
  z          Input
  3          Of length 3
 .:          Substrings. Without second input finds all substrings which is ok
  G          Lowercase aphabet

05AB1E, 7 6 5 bytes

Code:

3ãAŒÃ

Explanation:

3ã      # Cartesian product × 3 with input
  AΠ   # All substrings of the alphabet
    Ã   # Setwise intersection

Uses the CP-1252 encoding. Try it online!