This elevator is quite complicated, I tried to make something and hopefully it will work without any problems:
As soon as the elevator is being called a teleport trigger above the elevator will be activated for a few seconds(as long as you don't jump at the upper level of the elevator it would not teleport you. Trigger will be disabled 1.5 second before reaching its destination to provide players the momentum' boost while jumping), moreover, both upper and bottom doors will be close as long as the elevator starts moving and will be re-open when its reaches its destination to avoid more players teamkilling players from above or getting crushed to death from beneath(upper doors would be open early than the bottom doors to let players reach the bottom before the doors closes). In addition, added teleport beneath the elevator, when the elevator start moving it activated 2 seconds later in case someone got stuck there somehow and can not get out.
Feel free to test it and give feedbacks : ).