A method of producing and fining glass includes monitoring a temperature of a molten glass bath contained within a fining chamber of a fining vessel and, based on the monitored temperature, controlling an amount of a sulfate chemical fining agent added into a glass melt contained within an interior reaction chamber of an upstream submerged combustion melter that feeds the fining vessel. The temperature of the molten glass bath may be determined within a temperature indication zone that encompasses a subsurface portion of the molten glass bath that lies adjacent to a floor of a housing of the fining vessel. By monitoring the temperature of the molten glass bath and controlling the amount of the sulfate chemical fining agent added to the glass melt of the submerged combustion melter, the wasteful use of the sulfate chemical fining agent can be minimized and the fining process rendered more efficient.