Yes; the Olympus engines used were derivatives of an existing military turbojet and had reheat from the beginning. It was used for takeoff and then to get through the high-drag transonic regime as quickly as possible to minimise fuel use. Once at cruising speed the engines operated 'dry'.
There was a longer-range 'Concorde B' designed, which was intended to have larger engines with no reheat at all (as well as larger wings and some other changes) but this was cancelled when it became apparent that it would never be profitable.
OK, I think I got it wrong. You're correct it always had afterburners. It could supercruise, which is flying without the afterburner at supersonic speeds. So I guess that would bring the actual number to 3.
There was a longer-range 'Concorde B' designed, which was intended to have larger engines with no reheat at all (as well as larger wings and some other changes) but this was cancelled when it became apparent that it would never be profitable.