Turning spindles isn't rocket science. You don't need the same kind of set-up accuracy that you would for turning pens. I would think you could cobble up a temp. tailstock out of some scrap wood to hold a live center or a dead centerwithout much difficulty.
The problem I see is having a design that will allow the use of a tool rest along the entire length of whatever kind of extension you build.
If you want to build your own, I would do an Internet search. Seems like I have seen some articles describing simple home-built lathes. This might you pointed in the right direction:
http://www.mimf.com/articles/lathe/index.htm
That being said, the lathe extensions for the JET mini are not very expensive and you might even find one on sale around Christmas at a local Rockler or Woodcraft store if you have one near you. Ultimately, you might have more time tied up in the design and construction of a home-built rig than what an extension would cost you.
]
