Sachin,

You don't say what your current drain is !

Your problem is that the battery volts is very close to the supply volts you need for the board. When the SMPS fails, you need to switch in 6V and still get 5 volts to your system. For that you will need an LDO regulator.

A relay will make an admirable low loss switch, held on by the SMPS to route SMPS power to the system, released as the SMPS fails to allow the battery to come up.

Steve