Why does John Reese always speak with such a low voice volume?

John Reese is probably a high-functioning psychopath. His tone of voice is an indicator.

Although other people might see him as a world-weary ex-soldier who has suffered through the horrors of war, he is probably one of those who cause the horror. He enlisted in the Army to avoid charges, served in Special Forces, and was recruited by the CIA as an assassin (black ops).

Such people can be very charming, as I know from personal experience. I have known two, one of whom was a pattern for Reese, and both spoke the same, even when describing the murders of hundreds of innocent civilians. I was most impressed with the matter-of-fact tone of voice, usually blaming the victims for their own demise. They were “just in the wrong place at the wrong time, getting in the way”.

Perhaps he feels remorse for his past actions, and the on-screen depiction is that of a man looking for redemption, but I don't buy into that.

Several episodes comment on John's was of speaking, moving and interacting with others. Usually something is mentioned about his military background when the subject comes up. John's background is that he was special forces trained. He may have gotten into the habit of speaking quietly so as to communicate with team members on missions without allowing any opponents to detect his position.