How do I wire a new pull chain light from a preexisting switch for another light? I don't want the switch to control the new pull chain light. The switch should remain in place and continue to control the original light.

Find which wires are the feed at the switch and run from them. Use the same size wire.
If there is no feed in the box then it is not possible.

Check and see if there is power at the existing box with the light switch off. If no power you'll have to run more wire.

I'm curious Scott, with theswitch in the off position, how do you find power at the light.

Please read Stan's post. If the feed (source) is in the ceiling box, then you need to look in the box for the "Black" that feeds the switch and its white conductor in the cable. If the switch loop is wired by code, you should see a white conductor (feed to switch) connected to the supply black. Connect your new cable's black to this black/white connection point. Then connect the new white to the white that is paired to the source black.

Thanks everyone for the quick replies, but I'm still confused. The old switch has a black (OB) and a white (OW/B) wire with black tape on it. Can I use those wires? If yes, do I connect both the new black (NB) and the new white (NW) to the OB? Can I do that at the same splice?

If you are saying to connect the NB to the OB and the NW to the OW/B, won't the old fixture also turn on when I turn on the new pull chain light?

I also have a separate three-way switch to a different light in that switch box. If the above wires don't work can I use a wire(s) from the 3-way? Which ones? How?

No you can't use them, that's is a switch loop so you will need to pull the new wire from a ceiling box or wall outlet. One of the ceiling wires should show white with black tape too.

No, you cannot use the connections at the switch. What you have is a switch loop. All that is at the switch is power. There is no Neutral (return). Up in the geiling box, you should find a white with black tape on it. Connected to the black supply line. That is a retasked c onductor. It can only feed power to the switch.

The black from the switch returns the power to the light fixture's black.

The black supply line is spliced to the white ire (with black tape). You need to splice the new cable's black conductor to this connection point. ZThe white fron the new cable will connect to the hite from the supply cable.

At the new fixture's outlet box, the white will go to the fixture's white abd the black will connect to the black of the fixture.

Doing it this way will have the new light fixture work independent of the original light and switch