Right, an LDO is not suitable for you.
You have two choices, neither one very attractive. You can use a PNP or PMOS pass element with an active-regulator circuit that tracks the average value of Vin, but that's a painful circuit to make and leaves you with a brute-force compensated output. Or you can modify your circuit above to drive the NPN base from a voltage that's 400mV or so above Vin. To get this voltage you'll need to add a simple dc-dc converter with its output stacked on Vin, and appropriately regulate its output. For myself, often in this very situation, I settle for the modest drop across the pass element, and design the follow-on circuitry appropriately.
One way to improve the attenuation-vs-voltage-drop tradeoff is to make the filter active by splitting the base resistor and bypassing it from the output. The resulting 12dB/octave cutoff slope allows you to use lower-value resistors (less base-current voltage drop) and still get improved 120Hz and high-frequency noise attenuation.
. ,---||----------, . | | . Vin ---+------ | ---- C E ---+------ out . | | B . | | | . '-/\\/\\--+--/\\/\\--+---||--- gnd
An issue not always considered in these circuits is, what happens to the transistor's dissipation in the event of a short circuit? Unless Vin current limits at a fairly low current, the transistor may be exposed to a damaging power-dissipation level. I deal with this issue by adding a collector resistor, like a small 3W WW type power resistor. The resistor is chosen for less than 400mV drop at the maximum operating current. The tradeoffs in selecting this protection resistor reveal one more problem to solve in any attempt to design such a circuit with a voltage drop under about 700mV.
One last comment. If this type of filter is used directly after the 60Hz rectifier storage filter capacitor, where the ripple may be 500mV or more, the 0.5 Vbe average drop won't be high enough to allow proper filtering. In such a case, using a logic-level power MOSFET pass element not only gets you the extra voltage you need, but allows using much higher filter-resistor values, and eliminates the need for an awkward current-limit resistor, because the MOSFET delivers high output currents without excessive base-resistor drop.
Following a MOSFET filter stage with a three-terminal regulator is one way to get a very quiet high-current regulated power supply, with micro-volt ripple levels.
(One awkward issue you'll encounter is understanding the MOSFET's subthreshold region of operation to predict the Vgs value. Sadly, this issue won't be dealt with in the FET's datasheet. Moreover, available Spice models won't show the correct Vgs value either. But we do discuss the theory and give you guidance in AoE.)