I have never used a .308 at all on anything.
Anything 150 gr and up should do fine. Splurge on a premium bullet if you want. I've taken a number of moose with the 30-06, using 150, 165, and - my preference - 180 gr C&C. Some hand loads, some factory, some premiums.. Makes no difference. .... Or hasn't yet.
I've taken over 20 moose using the .243 (100gr), 12 guage slug, 270, 30-06, .338WM.
Makes no difference... As they say in the real-estate bidness," location, location, location".
The toughest one to go down was a spike yearling. He took the first one (Trophy Bonded), 250 gr .338 high through the shoulder blade , just under the spine, at 140 yards, the second took his knee out, the third as he was going away, just creasing the front of right ham, and exiting bullet #1 exit hole on far side after passing thru the paunch (uhg!), and the 4th - a 250 GK - same as first - finally put him down, tho he had only seconds left on his feet anyway at @ 160 yards. (1, 3,and 4 made one exit hole)
But he had his adrenaline up prior to #1. And it's not easy shooting offhand at a moving target while standing on a rotting birch log 4 feet off the ground, through a screen of alders. That's my story and I'm sticking to it.
I shot another yearling running, at about 35 yards with a 12 ga slug, neatly centering the big artery just under his backbone. He made about 100 yards before tipping over mid-stride, all bled out.
My wife double lunged a 3 year old bull at 80 yards with a 100 gr .243 Corelokt. Missed ribs on both sides, penciled right thru, turned the lungs to mush. He walked about 30 feet, stopped behind a screen of willows, stood there for a minute or so and tipped over.
'Tain't so much WHAT you hit 'em with..... Me - in heavy cover, especially, I've gone to CNS placement if I can get it.
"Good luck on finding consistent results on moose due to "caliber." : MD
