I suggest a MAX631, which is a switching regulator, with 80%+ efficiency. Since it is also a step-up regulator, even if the input voltage drops below the output voltage, it can still provide the output.
It requires just 2 external components + a current divider if you want to adjust the output voltage (which is your case, since the MAX631's predefined output is 5V)
I am very found of Maxim's voltage regulators. They have an extremely low dropout voltage.
I often use the MAX604 and MAX603 in my designs.
I think this would be good if the OP was using say 2 or 3AA batteries for the supply, which might be a good idea. Unfortunately the OP is using 9V, this would be extremely bad news for the load connected to the regulator as the output voltage will be 9V - 1 diode drop. The OP needs a buck regulator or to change the supply and use this regulator. I like the idea of 3AA cells though.