The most common and most likely answer to this crossword clue is the 4 letter word NARD
N
A
R
D
More clues leading to the result NARD
Buddy explains!

The straight part
We know that "Something oily" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case something oily = nard.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 NARD mean?
NARD noun- Nardostachys jatamansi, a flowering plant of the valerian family that grows in the Himalayas, used as a perfume, an incense, a sedative, and an herbal medicine.
- A fragrant oil from the plant, formerly much prized.
- American spikenard (Aralia racemosa), a North American perennial herb with an aromatic root.
- (1980s, usually plural) Testicles.