A steam generator has a heat exchange chamber with an upstream end and a downstream end. The steam generator further has a burner that injects primary reactants, including fuel and combustion air, into the chamber at the upstream end. A first branch of a once-through water line is located within the chamber downstream of the burner. A second branch of the once-through water line is connected in parallel with the first branch, and is located within the chamber downstream of the first branch. A fuel injector is arranged to inject staged fuel into the chamber at a staged location downstream of the first branch of the once-through water line.