I have a newer nylon-coated J Dewey cleaning rod in the 24" length used for my 16" AR.

I wasn't planning on it but recently picked up a 24" barreled bolt action and would like to use the same rod.

Does anyone offer just a simple 12" nylon-coated cleaning rod extension?

I just want some extra length so I can use the same rod on both guns. Plus my case is not big enough for a single 36" rod (both guns fit but my bolt action sticks out the top).

Thanks.

Not that I know of.  I just bought multiple different lengths myself to accommodate my different length uppers.

I've taken segments from my old brass 3 piece rod and added them to my carbon (Tipton) rods.  Not coated, but the brass is pretty safe.  I think if people are using coated rods they aren't going to want an extra threaded connection, so unlikely anyone makes a segmented rod that is coated or carbon fibre.
Subsequently I've bought different, longer rods so I don't have to add segments.

Thanks, I'll try that temporarily until I buy the 36" version. I had one brass rod from an Otis kit but it appears to be different threading and I didn't want to force it. See if I can find a different one.

I think you will have to buy a second rod.  If need a rod to fit the case, I'd put a brass segment in with your 24" rod and keep the 36" at home.  I'm assuming you want it to fit the case for portability/taking it to the range?