You could also place a diode on the regulator's input side to prevent the input LED from lighting up from any leakage currents. Just be sure to move the input cap to the regulator pin side. The TI version [(LM1117) datasheet][1] shows a similar example when using two regulators in a battery backup arrangement, (see figure 9-7).


  [1]: https://www.ti.com/lit/ds/symlink/lm1117.pdf?HQS=dis-dk-null-digikeymode-dsf-pf-null-wwe&ts=1686808388157