Pipevine relocating
 

I'd be inclined to do it in the spring before it leafs out but after
the worst of winter.
You can also root cuttings of it or maybe even mound up some loose soil
to layer it and make many new vines.

Pipevine relocating
 

I try to get a few nodes of green with a piece of woody stock on the
bottom.
I remove all but the topmost leaf and root my cuttings in moist
pearlite.