A bit of a random aside, but if you don't want to use poison / are afraid of secondary poisoning, you can make bait balls with peanut butter (or whatever food your mice are very excited about; bait balls made with poultry food mush and a binder work well if that's what they are already looking for) and baking soda. Rodents can't burp or vomit, much like rabbits and horses, so the Co2 will build up in their systems and kill them relatively quickly. Other animals, like dogs or other wildlife that decide to sample the bait or the dead mice will be unaffected.
Some will learn to associate the smell with "poison", but that is true of any bait block (and seems to be less likely with Co2 vs "regular" warfarin and other baits, but that's anecdotal)
It's not as clean of a death as cervical location by any means, but if someone is going to be using poison anyway that's probably not a huge concern for them. I try to share that alternative when poisoning comes up, 'cause while I agree it still sucks for the mouse, it sucks a lot less if a pet gets ahold of the trap or dead mice.