Basically, you need to move the shoulder forward without stretching the case too much.
There are a few ways to do that.
The difficult one is to make a "false shoulder" on the case neck by first expanding the neck to a larger caliber and then resizing it with the '06 die set to the proper headspace. That leaves an enlarged section of the neck to hold the case back against the breach when the cartridge is chambered, and prevents the case from stretching when fired. When fired with a normal load, the shoulder blows-out to the proper position. This is one method often used by "wildcatters" to make brass for a cartridge that they are developing.
The easy way is sometimes frowned-upon, because it does leave the possibility of stretching the brass too much. That is to make a low-powered load that is sufficient to "blow the shoulder forward," but not so high powered that it makes the brass grab the chamber wall tighly enough so that the case can't slip back without stretching. So, actually, instead of the shoulder going forward, the back of the neck expands to make new shoulder material. The case should actually get shorter instead of longer. Oiling the outside of the cases will help with that, but often isn't necessary. Figuring-out the proper load may require some trial-and-error unless you can get some advice here on how others have succeeded. I don't shoot '06s, so I will have to leave that to others. However, I will tell you that when I make .30 Herrett cases for my Contender, I use a mid-powered load with no lube on the cases to fully expand the case bodies after their initial forming. (I am not moving the shoulder location, but I am drawing brass from the necks to fill-out the case bodies. That typically causes the cases to get about 0.005" SHORTER. However, when I use full power loads in the same gun, those cases will get a few thousandths LONGER on each reloading, mainly because the Contender frame is much more "springy" than a bolt action, and the cases actually do stretch with those loads.
Finally, if the headspace error is not too large, some here may suggest that you can simply fire the brass and stretch it a little without causing an IMMEDIATE safety problem. Whether that causes a problem later is still a matter for consideration. If your brass is typically thrown away at its "end-of-life" because it is reaching incipient head separation, then this last approach would make that happen with fewer than usual reloadings. On the other hand, if you typically throw your brass away because of neck splits, then this technique MIGHT make the incipient head seperation become the reason for discarding this particular set of brass some time in the future. So, if you don't typically check the insides of your fired cases for thinning rings, you probably should with these cases.
SL1