Soaking in solvents will get alot of it out. Especially if you change the stuff for clean and resoak. You have to get the deep down oil to the top so it can be absorbed. No other way about it.
But even that will leave some of the deepest oil still in the wood. That'll weep to the top sometimes days or weeks after to spoil a finish.
Whiting powder is messy but the only thing I've ever found to get the oil out.
I use lacquer thinner soak first for a few days. Then whiting powder treatments as needed. Sometimes weeks are needed.
I mix the whiting with alcohol for application. It drys a little slower than the usually advised acetone for the mix and is easier to coat the wood that way.
Besides the acetone isn't going to draw any amount of extra oil out of the wood over the use of alcohol (cheaper). It's the whiting powder that does the work. The oil you're after is deep in the wood, not on the surface.
It's long drawn out process for a deeply oil soaked stock, but well worth it for restoration of an otherwise nice piece.
Having plenty of other things to do while it's hanging with a painted on coating of the stuff is the best way to avoid going at it too soon.
I've never had a piece fail in use after removing the oil. If the wood is strong, you can see and feel it.
I've even used the oven cleaner method at times. It's good for the surface and immediate subsurface oil and grease. The often mentioned ugly colors the wood turns afterwards along with the damage are easily avoided if you treat the wood with wood-bleach(oxalic acid) after flushing the cleaner.
The mild acidic wood bleach (don't use laundry bleach!)takes care of the base cleaner and leaves the wood natural color and very clean.
No stocks I've ever used it on have collapsed from dissintegrating cell structure as I've been warned.
It just never gets that far into the wood anyway. The lye in it turns oil to soap,,that's how it works.
But it won't get the deep down oil,,that's for the whiting powder.
With that said,,the 'green' cleaners now available (Purple Power(?) ect) do as good of a job and it's really easier & safer on you to use them for the first cleanup on the surface..
