Maybe
You did not say what brand of Rifle this was. If is a Ruger, or some other brand that uses a hammer forged stainless barrel, good luck.
The trick with any factory barrel is to get the thing set up in a lathe. Factory barrels are notorious for being crooked, even where the chamber enters the rest of the barrel. Just figuring out the best indicating points can be an exercise in futility.
Also, since the chamber is already reamed, your reamer pilot will not come into contact with the barrel ID untill it advances up in there quite a ways. That WSM reamer is quite a bit larger, you will soon see the problem.
Here is what I would do. Get the barrel chucked up as true as possible. Then, place the reamer at the edge of the chamber, and place a live center in the other end. (this will give two point support). Gently turn the reamer with a wrench, slowly feeding it with the tailstock, untill it starts to cut it's full diameter. Then just ream the chamber, (carefully), using this hand method, the reamer will follow the old chamber, or at least as good as could be expected, as you gentally push it in with the tailstock as you turn. Sooner or later, the pilot will start to engage the lands, and you are home free.
If this is a custom barrel, there is a good chance that the 308 chamber is pretty straight with the rest of the barrel. In that case, chuck up the barrel true, and rough bore the chamber to where the reamer will stick in far enough so that the pilot will engage the lands. Then, just ream in your normal fashion.........jackie
Just take your time, use plenty of cutting oil, and be careful.
