Howdy!
There are many ways to accomplish your goal.
First, you must determine if you want the finish IN the wood or ON the wood.
Old-time oil finishes were allowed to penetrate into the wood, and fill the pores, while newer finishes are more likely to be an external finish.
If your going for the new finish, the tint can be easily applied as the top coat, since it will not penetrate into the wood to any great extent.
However, if you are going the more traditional route, with an oil finish, as you seem to have indicated, the tint should be either applied to the bare wood before oiling, as a stain, or in the form of a tinted finishing oil.
I would definitely take the stock off the rifle, as you can get a lot of steel wool bits into the action even if careful, plus, it's almost, if not impossible, not to get finish onto the barrel and action, and between it and the stock, which can harden up like glue later, and make disassembly difficult.
Since you've already started the process, you must ask yourself if you want to start over.
If not, one solution would be to tint your own oil (artist supply houses, like Hobby Lobby have tinting agents) or hit the net and find some pre-64 Winchester finishing oil (I've recently seen it on ebay, and it was once called "red oil"), which has a slight reddish tint to it. This can be applied many ways, but, since it aint cheap, it's wise to rub it into the stock sparingly. It doesn't take much (except elbow grease, of course!), but expect to apply many coats, then cut them back off (The old timers, and Roy Dunlap, used raw linseed oil, cut with burlap, ACROSS THE GRAIN, to cut off the first coats and fill the pores) this is VERY time consuming and HARD WORK. The wood pores, if done this way properly, will be filled, forever, and the finish, in the eyes of traditionalists (like me) will be part of the wood, rather than a superficial coating. It will weather well and is easily touched up. (#0000 steel wool will work for the cutting off process, but can leave scratches)
By the way: commercial paint store BOILED LINSEED OIL is NOT the best type of oil for gunstocks. Artist supply houses (HOBBY LOBBY again) has much more highly refined oils which work better. Don't fall into the trap of adding "driers" to speed up the process, as some are toxic, and won't do what you probably want anyway.
A quicker method, which works fine with many folks, is to use "tung oil" to fill the pores because it dries quickly and can reduce the number of coats. One problem with tung oil is that a measurable percentage of people are allergic to it, and will be affected when they snuggle that warm stock up to their cheek.
Some apply a few coats of linseed (or even the proprietary :Linspeed Oil"), by rubbing them onto the tung oil surface, with high pressure, as this will cause the finish to SHINE!!
The English method, supposedly used at Purdy, was to soak the stock in hot oil, ALLOW IT TO THICKEN, THEN cut it off with burlap, check on pore filling, then do it again(and again and again,etc,,,) if needed, until it suited the maker. This took weeks.
One more thing: any wood will look good if you prepare it properly first. That means using the proper methods to give the wood that last touch, BEFORE the finish is applied. Some finishers stop sanding with #600 wet or dry sandpaper, but, take my advice and visit an auto paint supply house and introduce yourself to #1000, #1500, #2000 and #2500 grit sandpapers, if you want a truly sensual experience when you feel the sanded wood.
Fingers, rubbed across wood, sanded with #600 grit, can be HEARD.
Fingers, rubbed across wood, sanded with #2500 grit are QUIET, because it's so much smoother, and creates less friction. You WILL notice the difference, and nothing will ever be the same, when it comes to finishing.
The tiny scratches, from #600, will eventually resurface, sometimes years later, especially if the pores are not filled properly (which they seldom are), but the #2500 grit is FAR smoother, because of tinier scratches.
Finishing is FUN, but it is also very time consuming.
The good thing is that the principles are very simple
It's just a matter of patience, and more patience, and,,,,,,,,,,,,,
Good Luck.
Thanks for your time.