You cant usually, if the BG is jailed, cant save anyone, lets go by the narrative way of this game, someone is about to get shot, you intercept by pushing that someone outta the way, scaring away the assailaints, you cant do this when locked up.
Though, the detective still can investigate, from what I remember, the narrative is more like a file he has on suspects and having them staked out/followed by policemen, finding their out their shenigans (more or less finding out the results and filing it in midgnight, meaning, he can do this all from jail still, gets out from the locker then). Hence why jailing is a real useful way to save the detective in this game.
Its bit complicated, but more or less like that when I heard this mentionned. Never came up in this forum, but other forum I played, remember only thing viable was that the BG couldnt save anyone. Might disregard that here unless one decides on what to do etc.
P.S Sorry, Alen, my last post from now on.