Have you ever had to put out a small animal?

For some reason, I find animal suffering completely unbearable (even more so than humans, don't know why, I must be psychopatic or messed up in the head).

Just this week, I found a rather badly wounded bird and another rather badly wounded squirrel.

I wondered if I shouldn't just kill them to put them out of their suffering.
I mean, there was no way they would survive the night, I couldn't see the point of leaving them to suffer for hours and hours.

But I didn't quite know how to do it, as stupid as it sounds.
I ended up just moving them out of sight, hidden under some trees and let nature do what it's always done.
But I felt bad about it.

I thought about, I don't know, using a machete, axe or shovel, but I did not want to miss my shot have have to try again.
I thought about digging a small hole and squashing them with a huge rock.
Or drowning, but it's not instant and isn't fun for anybody.

So well, yeah.
How do you do it?
Do you even do it, or do you let nature run its course?