What do you do with the small piece of twisted pair cable jacket when you remove it, exposing the 4 pairs to prep for terminating?

Don't throw it away! It's a free tool & handy too. You can use it to separate the individual wire pairs by simply pushing and turning at the same time. Be sure to turn in the opposite direction of the twist and open up each pair end down to the cable jacket. For a professional version of your free tool, separator, take your scissors and cut jacket at an angle!