You cannot size 308 brass with a 30-06 die because the die will bottom out before the case neck is sized.
You CAN neck size 30-06 brass in a CONVENTIONAL 308 die. But it seems apparent that you have a collet die, not a conventional die.
If you still believe you have a conventional die, simply remove the expander and size a case. If you can insert a bullet into the neck, then either the sizer is bad or your brass has unusually thin necks. You shouldn't have thin necks with once-fired brass, but with brass of unknown origin there's a possibility the case necks have been reamed or turned to accommodate an undersize chamber. (Turning case necks is SOP for benchrest shooters.) I find it hard to believe your sizing die is bad, but after all we ARE talking about a Lee "Precision" product.
Just so we're on the same page, a conventional sizing die sizes the case neck in two operations: During the downstroke, the sizer enters the oversize case neck, which is then squeezed undersize by the die body. Then, on the upstroke the expander is pulled through the now undersize case neck, expanding it to the appropriate i.d.
A collet die sizes by squeezing the case neck between the body and mandrel. Naturally the case neck will spring back after sizing, so the mandrel has to be a bit undersize.
It seems you're not the only one to have problems with the Lee Wonder T*rd Collet Die; see the link below: