Good morning Simon...see if you can make sense of this.
Because chain link is quite stretchy, the following is easy to do. I just relocated the old fence about 12" into the neighbours place, using a few steel fence posts...didn't need to unfasten either end as they are dealt with last. I ran a string line along the proposed permanent position and placed my new posts in the correct position and spacing. Once I was satisfied with all that, I ran out my new wire for the full length. I fastened at one end and gradually moved along the line, standing up the chain mesh, tightening to about the correct tension and holding it in place with the cable ties...once you have it all standing for the full length and held up safely, go back and tighten each section to how you want it...this will work quickly and easily if you have a helper or two, just to fasten the wire to the posts while you hold the tension on.
Once you are satisfied with the result, go next door and remove the old fence, roll it up and if you aren't planning on using the wire any more, place it out on the side of the road...it will be gone in a flash!!
Hope that helps make the job easier for you.
PS. I didn't have a helper, but I do have a smallish hand winch, so once the first end was fastened properly, I just hooked up to the other end and tightened.