The most common and most likely answer to this crossword clue is the 8 letter word UNPAIRED
I've seen this crossword clue in The Daily Telegraph
U
N
P
A
I
R
E
D
More clues leading to the result UNPAIRED
Buddy explains!

The straight part
We know that "Odd" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case odd = unpaired.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does UNPAIRED mean?
UNPAIRED verb- To go from a paired to a non-paired state; to disassociate.
- Not forming one of a pair