Very experienced Goat hunter, passed on this information Goats are tall and not that wide, lots of people try to use bullets that are to hard, plus long hair is misleading on shot placement. Need penetration and enough expansion to shock them.

as stated elsewhere, its bullet placement that is important. I used a 300 Win Mag, with 180 grain Nosler Partition. Shot him through the shoulder dead in his bed

300 Win Mag shooting the Nosler 180gr Partition worked well for me.

I don't really have a big total of animals killed with my 338 win mag. but all of them have been shot with a Barnes 210 Grain TSX. All DRT. If you shoot through the shoulder I don't think there is any critter that won't go down.
I think the 338-06 is a great choice as long as your ranges are reasonable.

I've never shot a goat but if I were hunting them with a 330/06 I'd load up some 210 grain Nosler Partitions and head to the mountains without any concern about the bullet. I have shot a whole bunch of African plains game from Impala to Eland with the 225 Gr. Nosler Partition in my .338 Win Mag with great results.

I'm a big fan of the Partitions.

Another vote for Nosler partition. The right combination of penetration and shock. Big, old billies are tough, but some of their "bullet absorbing" reputation comes from the fact that when a goat walks 10 feet after the shot there might be a disaster, but when a moose walks 10 feet and drops we say, "he was killed instantly". In other words, it has a lot to do with the terrain in which you hunt goats.