The
TI TPS7A20 datasheet recommends 1µF input capacitor and 1µF–200µF output capacitor with less than 0.1Ω ESR (C0G, X7R, or X5R type).
It seems to me –– but I'm only a hobbyist myself –– that say 10µF on output, and 1µF and a ferrite bead on input, would work well. Datasheet figure 6-64 shows that using 10µF on the output helps better power supply rejection ratio (PSRR, ie. rejection of supply noise to the input side of the regulator) around 1 or 2 MHz.
I usually power my own circuits from USB 5V, and that has a limit of maximum 10µF capacitance; so, I tend to use regulators that tolerate sufficient capacitance on the output to give me stable voltages. For ADCs and similar having their own supplies, I like to use a ferrite bead to absorb/reflect the high-frequency noise (say, above 10 MHz).
In this particular case, doing a quick
.ac oct 100 1K 1MEG simulation in KiCAD/ngspice, simulating the latter filter with a 10Ω load (for 300mA current at 3V,
VSIN dc=3 ampl=0.1 f=1k ac=1), shows 16.7 dBV gain (6.8× amplification) peak at 23 kHz. It only becomes worse with lesser output loads: at 3kΩ (1mA load at 3V), the gain is over 47.3 dBV (230× amplification). This makes the filter
amplify noise between 5 kHz and 35 kHz or so, especially at low loads. Now, I could easily be wrong here, so I hope more experienced members will pipe up, but to me, it looks like maybe the Pi filter is not what one actually wants or needs here.