I have an existing pull chain light fixture installed in an area in the basement. I need more light in that area. I want to add a second fixture in the area. Can I connect it to the existing pull chain fixture in some way so that the new fixture will operate off the existing fixture and turn on (and off) in tandem with the existing fixture? Or do I need to wire it as a separate light with it's own pull chain?

No, but a qualified no.

You can add a light but you cannot make it turn on and off with the pull chain of the first light.

The only way would be to access the inside of the switch and wire the second light from there.

You could put in a twin light adapter and screw in a second bulb.
Or you could put a bulb in one side of the twin light adapter and then put a plug adapter in the other side. Cut off the female end of an extension cord and attach it to a second bulb holder and plug it into the plug adapter in the first bulb holder.

Actually I believe you can do what you want. Watch this video and tell us if you can get to the black wire AFTER the switch. https://www.youtube.com/watch?v=k4XBu6Z81rI

Thanks for confirming I cannot (easily) wire the second light fixture to the first. I had wired it several different ways with no success but had hoped for another option. I appreciate your suggestions - those are very helpful.

If you need more light, might I suggest you replace the bulb with a led bulb. I needed to replace a standard kitchen bulb which was a recessed 75 watt bulb. When I put in a replacement led bulb. I go so much light that I don't even have to turn on my light over the kitchen table anymore.. There are 2 types one is switchable and non switchable. I chose the non as that is all that I needed. It will take some getting use to the led light as it is very bright but not pure white light. Just a suggestion to improve your lighting without any real changes. LED's claim 22 years life and cheap as dirt electricity.

Since pull chain lights in a basements are now not permitted, unless right at the door, the simplest thing is to just install a wall switch with either hot through or switch looped and add the second light. I do a lot of work in older homes and are always eliminating pull chain lights. In fact, you can leave the pull chain fixture in the ON position and wire additional lights. Typical comment "I've walked in the dark basement for --years, now I can see".

Since pull chain lights in a basements are now not permitted, unless right at the door
When did that happen Brian?

My last NEC is 2011. In 210.70 A3- basements.. at least one lighting outlet containing a switch or controlled by a wall switch... The point being that you shouldn't have to walk through a dark room to find a pull chain. Somewhere else it indicates.. at point of entry. So, at least in my take, you can have a pull chain luminaire if the pull chain is at the doorway but not in the middle of the room. I see a lot of inspection notes regarding home sales and this is commonly noted. I'll make reference next time I see one.

My last NEC is 2011. In 210.70 A3- basements.. at least one lighting outlet containing a switch or controlled by a wall switch.
That doesn't say that there can't be a pull chain light. That only says that there must be one switch controlled light.

I realize you are not quoting directly from code and I do not have current NEC code book.

PS
I was in Home Depot earlier this evening. Look for a bulb holder. Couldn't find them, kept going up and down the aisles looking (they were out). Happen to notice the 14/2 Romex. It was $32.97 for 250 ft. roll. Its been a long time since i bought any 14-2 and I believe it was up around $60 to $70. What happened?

My last NEC is 2011. In 210.70 A3- basements.. at least one lighting outlet containing a switch or controlled by a wall switch

Can you accept that a pull chain is a "lighting outlet containing a switch"?

that being said, I agree putting a switch at the entry is preferred!

The basement listing is a bit more vague than living areas. However, the idea is you shouldn't have to walk into a dark room to find a pull chain. My newer house, 8 yrs. old, has a shop, 2 storage rooms, a service room and other rooms. All rooms had wall switches connected to a ceiling porcelain light fixture.
Here is a living area synopsis

Living Room, Dining Room, and Bed Rooms
These rooms require that a wall switch be placed beside the entry door of the room so that you can light the room before entering it. It can either control a ceiling light, a wall light, or an outlet connected to a lamp The ceiling fixture must be controlled by a wall switch and not a pull chain type light.