A method is disclosed of providing a fuel efficient regeneration of an exhaust after-treatment (AT) system that includes a lean oxides of nitrogen (NOX) trap (LNT) and a selective catalytic reduction filter (SCRF) positioned downstream of the LNT. The method includes regulating a selectable position valve. The valve permits a first gas flow portion to pass through the LNT and diverts a remaining second portion of exhaust gas flow from a first passage connecting an engine and the AT system to a second exhaust passage to thereby bypass the LNT. The method also includes regulating a first device to inject fuel into the first portion of the exhaust gas flow. The injection of fuel in to the first portion of the exhaust gas flow provides fuel efficient regeneration of the LNT and promotes NOX conversion and ammonia (NH3) formation in the LNT. A system employing the method is also disclosed.