If it's truly an oil finish and not a dull varnish "fake oil" finish then the correct answer is "more oil". For an actual oil finish you can forget the ScratchX. That's only useful on varnish or lacquer finishes for polishing them up.
Your description of the wood "bleaching" out sounds like the oil finish is being washed out and the wood is becoming dry looking and dull appearing. That's a classic sign for an oiled finish which is being flushed away and oxidized by the elements so I tend to believe that yo've got an actual oil finish without having to see it.
With an oil finish the right course of action is to find out which oil Sako suggests as a feeding and recoating treatment and apply a light coating, allow to soak in for a minute or two then buff off the excess.
The thing with a true oil finish is that it's never finished. There's an old wood finishers story about doing oil finishes. It's "Once a day for a week, once a week for a month, once a month for a year, and once a year forever". This recognizes that the oil finishes wear and flush away from use and exposure. So it's actually essential that the owner of such a rifle should recoat with the same or some compatible oil when the wood is looking doggy.
I've been using Lee Valley's polymerized tung oil finish on some refinished old gun stocks for use in my cowboy shooting and on some custom pistol grips. It's standing up well to wet weather and sweaty hands in all cases. It's a great finish for wood being used in a hunting situation where the conditions are often wet. If your Sako is finished with tung oil then this would be a good option. If your stock is REALLY doggy and dull then two coatings applied and buffed off about a week apart when the smell of the first coat is all but gone would not be a bad option. You want to get it to where the finish has a bit of a "glow" and shine to it. With that level of build the wood will have a really nice glow and clarity to it and water will bead and run off like off a duck's back.
The ScratchX treatment is popular with some owners of rifles that have a matt finish varnish. It makes them shiney like they were waxed up nicely. But on a true oil finish it'll just plug up the pores in the wood and look ugly.