The solution to this would be simple for me. I have always made sure that the bullet was seated to a min depth of one bullet caliber. The idea as I learned it was it needed that much to hold the bullet firmly. What I found is that nearly all heavy bullet's in my rifle's, generally would make that, some ended up seating to the base of the neck. One rifle, a Rem 700 with a 100gr bullet, I couldn't get close to the lands. Problem was the magazine well was to short, I lived with it and it was fine. 100gr bullet's was all I'd ever tried in that rifle. In an old L61R Sako in 7mm mag years ago. using 160gr bullet's, the bullet went below the neck and was just off the lands. Plenty of magazine so I made up a dummy with the bullet seated to the base of the necf and had a gunsmith lengthen the chamber so that round fit just off the lands, worked like a charm.
You pretty much have to live with what you have. It sounds like you have a really long chamber and to hold the bullet at one caliber depth, your gonna have to let the bullet jump quite a bit. Weathery's all have a bunch of jump and I guess some of them still shoot well. You really need to get a handle on how long your cartridge can be. Try this, close the bolt on an MTY chamber and stick a cleaning rod down the barrel to the face of the bolt and mark the rod right at the barrel. Remove the bolt and drop a bullet into the chamber. get a new wood pencil and push the bullet until the lands stop it. Run the cleaning rod down the barrel again until the nose of the bullet stops the rod, hold the bullet in all the time. Mark the rod again and then measure the distance between the marks with your caliper. That is the length from the bolt face to the bullet tip. Make a dummy round that length, OLL. It will not chamber without hitting he lands. Seat just a bit more until the bullet no longer touch's the lands. Watch the land marks on the bullet as you go and every time you seat it deeper clean the marks by rubbing off with 0000 steel wool. Keep doing it till land mars no longer show at all. Even after you get to where you can't feel them, I've found they still show faintly, not good enough, go just a bit deeper.
Now, if your chamber is simply to long, seat the bullet to one cal depth. With the 308 the bullet will be seated to where the neck and shoulder meet. That will leave you some bullet jump, the round should go right into the rifle and chamber. Load up some rounds and shoot them and see if your rifle can live with it.
If you still don't want that jump, take a dummy round to a gunsmith and have him cut the chamber to fit the load. If you get good results after seating to the base of the neck, I'd live with it myself. I would strongly suggest you do not seat the bullet as shallow as you say and try it. The bullet there could be seated off center and you may not even notice it.