For the achievement, you have to complete any 10 Rescue missions.
You really want to have Loki with invisibility. If you do, then there are several low level Rescue missions to choose from:Grineer
-- Spear Sector
-- one of few low level Rescue missions that is Grineer. Watch out for energy sapping doors!Corpus
-- Linea SectorVenus
-- V Prime Sector
Unfortunately, setting the hostage free does mean you have to run all the way to extraction to count for the achievement. As long as the alarm doesn't go off in the Hostage area before you find the Hostage though, you're good!
However, you don't really have to watch the Hostage on the way out. Run as fast as you can and he usually teleports to you quite often!
I've included my loadouts for speed or invisibility on my Loki Prime below; I personally liked the ability to move very fast just to get to the objectives and extract faster.
My Loki Prime max speed build(Sprint speed 1.96):
My Loki Prime invisibility build(203% Power Duration):