"a pipe in the hallway leading seemingly to nowhere" - if it is a single pipe, how did you propose to fit a rad? - you need 2 pipes - a feed and a return.
It is probably getting warm because it is connected to the existing system and heat flows down the pipe in both the copper and the water inside it through conduction.
If you (or partner) have never fitted CH before, I don't recommend you start here. You will invariably have to do some pipe cutting and attaching valves to the ends.