I talked to my sister a few days ago on the phone and asked her for numbers, and I asked her how many times her 308 bullets exited her moose.
As near as she can remember she has used her M88 to kill "about 25" and also "about 15 Caribou and 1 grizzly". She used a 30-06 on a few moose also, and 2 she killed with a 338. She doesn't like the 338 because it kicks her too hard and she's tall but thin.
She said about 1/2 of the kills with her 308 gave her exits on moose, and all her caribou had exits except 1. I loaded all her 308 ammo and used 180 grain Nosler Partitions, and 41.4 gr of 4064, so 100% of those animals were killed with that load. She said shots on all her moose were from 20 yards to about 150. Never needed to shoot farther on a moose, or her bear either.
I was only with her on one kill in the last 30 years, (and for that kill she used her husband's 30-06 with a 200 grain Nosler Partition) so I just have to go on the information she tells me, but I just loaded another batch of ammo for her rifle and my Nephew picked it up here to take it back, just about 2 months ago. She is very happy using her Winchester 88 and the ammo I loaded for her before (in 1989) so she didn't want to change anything.
Her husband had a 30-06 Mauser with a Leupold 4X scope and a Winchester M70 in 338 with the same kind of scope as he has on the 30-06 but her M88 has a peep sight I installed on it just before she moved to Alaska in 89. Moose are big, and she's never wanted to scope her 308 because she shoots well, and all her moose have been killed under 150 yards. She likes being able to hold her rifle with no scope in the way, and likes how slender it feels compared to the scoped rifles. As we age however I would not be surprised to see her cave in on the scope in the coming years. She and I both have noticed that dim light is a problem now, where when we were younger it didn't seem to matter as much. She makes a point of shooting them through the lungs because they are both hunting for meat, and trophy size is not very important to them. As a rule she shoots and (to quote her) "I watch the moose fall down in about 10 seconds or less".
Personally I have killed only 4 moose in my life and of the 4, 3 were the same, falling after the shot 10-20 seconds with double lung shots. 2 with my 375H&H and 1 with a 348 Winchester. Not one of those 3 fell at the shot, and all of then were hit perfectly.
The one that fell instantly was hit with a flintlock in 62 caliber with a round ball shot from about 80 yards. That one was hit in the shoulders instead of just through the ribs, but I was quite impressed at how well that big ball put the bull down.
And as a side note, the only bullet I recovered in one piece was the one from my 348, a 200 grain Hornady. I recovered the jacket from my Sierra 300 grain 375, but I still got an exit, so I have to say at least part of the core punched through. I shot one bull with two 285 Grain Speer Grand Slams and both exited but as far as I can tell, neither expanded. One hit a rib on it's way out and didn't even cut it completely off. So I have to assume they were made for much thicker skinned game then american moose.
Flap Flap Flap................. but rifle loonies will probably like the info. Safe to say to the OP, if you load a good 180 grain bullet that will hold together you will not have any problems killing a moose with your 308. Lighter ones will kill too, but I expect less penetration with less weight and from what you said and what my sister has done for over 30 seasons you are likely to shoot close up, so faster speed is not going to help you at all, but heavier bullets may.
