Before you can Skulljack for non-plot reasons you first need to research "Skullmine" in the proving grounds. It only works on the Advent trooper family of enemies, not on pure-breed aliens.
Successfully hitting an enemy in melee with the skulljack gives you a hacking attempt with a small reward of a small supply bonus and a large reward of a lead on an alien facility. Failing the hack deals 3-4 damage to the x-com operative. The success chance of non-specialists is very low, so you should leave skulljacking to the specialist class.
As a secondary benefit, a skulljack-hit is always an instant kill, even when the hacking is failed. So sometimes you might want to do one simply for tactical reasons.
The "special security feature" which you encountered during your first hack into the advent network will not bother you during non-story-relevant skulljacks.