This achievement description is a little strange and doesn’t entirely give you the parameters you need to complete it. This guide is the method I used to get the achievement.
Ready or Not Way Out West Achievement Guide
This can be done on either Commander or Practice mode. “Coyote” may have been the old name of the map, but the map you want to choose is now called Rust Belt (Costa Vino Border Reserve).
Though the achievement says to only use the .357 as your “weapon”, there’s some strange stipulations with what can be used to get the achievement unlocked. A.I. squad members can use whatever guns they like without ruining the run. However, there’s some reports that having them use equipment (flashes, C2, etc.) ruins the run, so avoid using those if possible. On your player character, bring your Magnum full of rounds and take away primary ammo as to not accidentally use it. I used a shield as well for the back protection, though some people say that taking it out and using it during the mission made it not unlock for them, so stick to having it on your back. Lastly I equipped the night vision goggles to make it easier to see in the tunnels, since the Magnum doesn’t have a flashlight attachment.
This achievement turns out to be a lot simpler than it may seem. During the entirety of the mission, only have the Magnum equipped and don’t switch off of it. Send your squad mates in to every room, avoiding the use of grenades or breaching tools if possible (again, varying accounts on if that affects it or not). They will be able to use their full arsenal without ruining your progress. If any suspects fire at you use the Magnum to take them down (it’s powerful enough to put most enemies down in one center-mass shot). Besides that, all you need to do is just walk behind your squad and let them do all of the heavy lifting. There isn’t a minimum grade requirement, but make sure you complete the hard objectives without killing civvies.
In short, command your men to clear all rooms, don’t use equipment, and complete the hard objectives and you’ll get it on your first run.