Find all the Anagrams, and the Subanagrams too!

Brachylog (2), 7 bytes

{sp}ᶠdb

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Explanation

{sp}ᶠdb
{  }ᶠ    Find all
  p        permutations of
 s         a substring of {the input}
     d   Remove duplicates (leaving the list otherwise in the same order)
      b  Remove the first (the input itself)

05AB1E, 7 bytes

Œ€œ˜Ù¹K

A function that accepts a string from input and leaves a list of strings on the stack. As a full program a representation of the list is printed.

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How?

        - push input
Œ       - all substrings
 €œ     - for €ach: all permutations
   ˜    - flatten
    Ù   - de-duplicate
     ¹  - push 1st input onto top of stack
      K - pop a,b; push a without any b's (remove the copy of the input from the list)
        - as a full program: implicit print of the top of the stack

Jelly, 9 bytes

ẆŒ!€;/QḟW

A monadic link accepting a list and returning a list of all distinct sub-anagrams except the input itself.

Try it online! (the footer pretty-prints the resulting list by joining with newlines.)

How?

ẆŒ!€;/QḟW - Link: list of characters, s
Ẇ         - all contiguous sublists of s
 Œ!€      - all permutations for €ach sublist now a list of lists of lists)
     /    - reduce by:
    ;     -   concatenation (flattens the list by one level)
      Q   - de-duplicate (gets the unique entries)
        W - wrap s in a list (otherwise filtering will attempt to remove characters)
       ḟ  - filter discard from left if in right (remove the one equal to the input)