I'm working on a circuit modification ... that will work for ... DC with 3 to 10 volts applied to the rails ... 5-volt LED strips instead of the typical 12-volt strips ...
What circuit are you modifying? In the circuits shown so far in this thread, the output voltage is LOWER than the input voltage. So if you only have 3V DC on the rails you can't generate 5V on the output. In fact, with the circuits shown the regulator IC device itself (whether the OP's 7809 or GRJ's LM317) needs an additional couple volts of so-called headroom or dropout voltage to perform its function. In other words you need, say, 7V going in if you want 5V coming out. Additionally the bridge rectifier itself drops the voltage by over 1V from the rail to the regulator IC.
If you have the space, you might consider a so-called buck-boost regulator module for about $2 on eBay. These have been described in various O-gauge passenger car lighting threads to drive 12V LED strips across the full-range of conventional/command O-gauge rail voltages (where the track voltage can be below OR above 12V).
