He used it during his time as a Time Agent, and beyond.
After Rose Tyler as the Bad Wolf resurrected Jack following his extermination by the Daleks, (TV: The Parting of the Ways) Jack used his manipulator to travel from the Game Station in 200,100. He attempted to reach 21st century Earth, but missed and ended up in 1869. The manipulator burned out after this trip. However, Jack was still able to use at least some of the other functions besides time travelling and teleporting. (TV: Utopia)
Years later, as Jack, the Tenth Doctor and Martha Jones were being attacked by Futurekind on Malcassairo in the year 100,000,000,000,000, the Doctor repaired the vortex manipulator with his sonic screwdriver, transporting them to Earth in 2008. The group used it to teleport to the Valiant, from which Martha subsequently escaped, also by use of the manipulator. (TV: The Sound of Drums) Martha considered the vortex manipulator in her Top Five Worst Ways To Travel because it made her sinuses and organs ache and caused her throat to become bloody. (PROSE: The Story of Martha) The Master later also used it to teleport from the Valiant to Earth, where he threatened to destroy the planet. The Doctor was able to use the manipulator to teleport them both back to the Valiant. Once the Master was defeated and Jack made his way back to the Torchwood Hub, the Doctor deactivated the manipulator, saying that he couldn't leave Jack with a time-travelling teleport. "You could go anywhere in time, twice; second time to apologise", according to the Doctor. (TV: Last of the Time Lords)
When the Daleks relocated Earth to the Medusa Cascade and invaded it, Jack used information from the UNIT Project Indigo (based on Sontaran teleport pod technology) to reactivate the teleport function on his vortex manipulator (he still couldn't time travel). He then teleported to find the Doctor, arriving just after the Doctor was shot by a Dalek, which Jack then killed. (TV: The Stolen Earth) After the Daleks were defeated, the Doctor deactivated the manipulator again, sternly saying "I told you - no teleport". (TV: Journey's End)
Eventually, Jack bequeathed his vortex manipulator to UNIT after one of his deaths. It was held in their Black Archive under the tightest secrecy, with UNIT keeping it hidden from not only its allies (particularly the Americans) but the Doctor himself. After the Zygon infiltration of UNIT was exposed, Clara Oswald took the manipulator and, using the activation code left by the Eleventh Doctor, used it to reunite with him in Elizabethan England. (TV: The Day of the Doctor)
Behind the scenes Edit
Jack's vortex manipulator was used much in the same way as the Doctor's sonic screwdriver in the context of Torchwood, being capable of doing just about whatever the writers wanted it to. Its only known limitation, within the context of Torchwood episodes, was being unable to time travel or teleport, both functions having been deactivated by the Tenth Doctor.