A method is disclosed for controlling addition of a supplementary oxygen stream into a steam methane reformer utilizing both the supplementary oxygen stream and a primary oxidant stream to support combustion of a fuel stream by burners firing into a radiant section of the reformer. The combustion generates heat to support endothermic heat requirements of the reforming reaction conducted in reformer tubes to obtain an enhanced rate of production of a product gas stream produced by the endothermic reaction. In the method, a temperature is obtained that is at least referable to a reformer tube wall temperature measured at a location of inlet regions of the reformer tubes at which a maximum temperature is produced at the enhanced rate of production. This temperature is controlled by regulating the flow rate of the supplementary oxygen stream to either prevent damage to the reformer tubes at such location at which the maximum temperature is produced or to maintain the maximum temperature if the same is less than a temperature that will damage the tubes.