Velocity is your friend, and 38 special doesn't have much of that. Just a fat 22 sub all up
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Still allowed to be a semi auto and you get more decent sized pellets in the spread.
If there were decent mobs and you're a good shot with appropriate ranges then you can deal with a few. As opposed to a tiny 22 mag or a bolt.
These were around 1000 fps. Not sure what the bullet size was. Some big cast ones. Thinking around 130 grain.
.357 mag lever gun gets the job done, .44 mag for that matter. I tend to find getting close to goats is a problem where I've gone, but 00 buck gets the job done within 30m with a tight choke. To be fair slugs are what I'd consider as they can be pretty accurate even out of a smooth barrel cylinder bore. My mossberg is reliably accurate with quality slugs. There are some Italian rounds called target slugs that are cheap and reasonably good about $35 for 25 that I buy from Wills Outfitters in Kumeu.
Id see whats available in semi or lever 22mag or ill thin them out for you
As you mentioned neighboring properties and safe back drops buck shot seems sensible. I have shot hundreds of pigs and goats with it. Just pattern your gun starting at 25 yards and go up from there, what ever range it still throws a decent pattern out to is the range you stick with. Sometimes modified choke throws better patterns than say full so try a couple of chokes.
Before mag extensions became a prohibited item one could sneak right into the middle of a mob of goats and often there wouldn't be many that got away.