I just read this thread for the first time and was surprised to see that no one uses the method that I use. I change three strings at a time on guitar. I fasten each string at the bridge with the pin, then wrap the string around the post three times with as little slack as possible, pull the string through the hole, then tighten the string to pitch. I can often replace a broken string before the song ends using this method.
When I'm putting on a new set of strings, I cut them so about 1/4 inch protrudes from the hole. If I'm in a rush - changing a broken string on stage - I curl the string using a loonie or a quarter. This is the same method that people use to curl ribbon for Christmas presents, using the blade of a pair of scissors. I clip the end as soon as I get a chance. I always try to have a string winder on hand.