Without seeing what you have, without knowing what type of finish it
is, without knowing what kind of sheen you are looking at, without
knowing if it is tinted one step clear spray finish or not, without
knowing how the finish is applied....
you are up against it.
While suggestions are good, they won't be much more than that. If it
is a valued piece, take to to a finish repair shop. If the oils
dissolved the finish (no matter what the finish is) then no doubt they
penetrated the wood.
To attempt to neutralize the effects of the remaining oil left on the
wood (penetrated?) to leave a bondable substrate for your finish while
matching an existing color is not for the inexperienced.
Good luck, and I mean that sincerely.