You're wanting to rechamber only.....keeping the barrel
In that case you can rechamber to 300winmag which is the same length as the '06 (by design)...
Which is a great way to launch a 30cal...and the 300WinMag round will fit in the mag and feed (may need minor work) will need to open up the bolt face to mag size
Or you can go hard (or go home) with a supermag.... but a bit more barrel might be more ideal for the greater powder
RUM will clean up the '06 chamber as well, same mag bolt face but length without mods would be a single feed..
300 Norma Mag (as opposed to 308NM) is a bit longer than the RUM in COAL but shorter case, so (I guess) designed for longer more stream lined bullets?? it has the larger rim size (338LapuaMag size) so that might be an issue.
30-378 similar to the above with rim, also length is long at 95+mm and barrel length issue...
You could even 'old skool' it with a H&H...similar to the win mag in performance but mag length cartridge like the RUM. So a skinny tapered RUM with a belt...