The 340 never came in 25-20, your thinking of the Model 23, I had one of those in 22H. The 340 came in 22H, 222 Rem, 223 Rem, 225 Win and 30-30 Win. As I mentioned earlier the 340 has only one action screw at the front of the receiver and the barrel band to hold it in the stock so it is impractical to try and free float the barrel.
I had seen an article about adding a second action screw to the back of the receiver and it is quite involved and would necessitate some serious metal and stock work. Here is the article with lots of info on the 340.
Personally, I would full length bed the barrel, action and keep the barrel band. Mine shot marvelously with that set up.
Have fun with your 222 Rem, I have four, one set up for varmints and three for calling coyotes, it is one of my go-to rifles for predator hunting. Now I prefer the Speer 52gr Flatbase HP for hunting in the 222. I won the PM Egg Shoot with a 222 Rem.