Most of the soak methods will get the surface and near surface oils.
But the deep down oil needs to come out too or it'll just make it's way to the surface sooner or later.
After years of trying numerous methods of getting oil out of gunstocks, primarily SxS butt stocks & forends, I've settled on a soak in laquer thinner first.
It does as good a job as acetone, doesn't evaporate as fast when working with it, hopefully isn't as bad for you,,but still does evaporate from the wood in a 1/2 hour or so (usually less) once pulled from the stuff.
Once out of the thinner soak, I use whiting powder. A can from Brownells has lasted a long time even though I use it all the time.
I use cheap rubbing alcohol to mix the paste up instead of acetone.
Acetone may cut the oil in the wood better but it evaporates so fast before it ever gets a chance to do that and that's not the role of the solvent in the paste anyway.
The alcohol (or acetone or laq thinner,,what ever you want to use in mixing the paste) is there only to make the whiting powder into the paste so you can get it onto the wood.
Reletively fast drying is a plus so it doesn't run, but it doesn't have to speed dry as acetone does.
It drys into a hard but brittle shell of whiting encasing & clinging to the wood and absorbs the oil as it makes it way to the surface.
As the coating turns yellow/brown after a day or so, brush it off and reapply another coating, remove that one when it turns color and reapply, again and again till no more oil discolors the wood.
It may take a couple weeks to a month to do a stock. But it is dry of oil when I'm done. Small price in time to pay when everything else done after that depends on an oil free wood to work on.
I sometimes have as many as a half dozen stocks and/or forends in the process at one time. Sometimes I forget about one or some of them but it doesn't matter. When I get to it, a quick brushing and recoat and back on the hook.
Yes it is a painfully slow process, but the only way you'll know that all the oil is removed (the whiting discoloration is the indicator). The other methods simply do not work to bring the deep down oil to the surface and extract & remove them in their shorter work times.
The oil can be several inches into the head of a butt stock/pistol grip and simply soaked thru side to side.
I've used the method for many years now on everything from Parkers to Purdeys to Military rifle restorations.
I'd used other methods like kitty litter, heat lamps, oil dry, water based solvents/cleaners, ect. But none do the complete job though they may seem too.
Leave the piece alone when you think it's dry of oil for a week and see if it starts to leak again.
If you don't want to harm existing finish, skip the laquer thinner soak and go right to the whiting. It'll just take a few more applications.