You got a couple options that run from selling the rod and buying a new rod that meets your needs to trying to modify the current rod.
I have never replaced a butt cork before but believe I understand how it us usually done without removing everything ahead of the butt cork (ie. reel seat, fore cork, eyes etc....)
Removing the cork is the easy part

adding back on a new cork is where it gets a little tougher. You need to adjust for the taper in the rod. This can be done by building the rod up into a straight shaft by a couple methods. I would wrap thread on the rod until I had removed the taper from the reel seat to the butt end of the rod. Masking tape wrapped tightly could do the same thing. Then you will need to lengthen the blank. This will be far tougher than any part of the project. I have never read any material on lengthening rods but can make a couple guesses at how to do it. You will need to find something that will join up to the existing blank and be able to transfer load from this added on piece to the existing blank. A wooden dowl may do this but I would try and find a piece of rod and epoxy it up into the existing blank. How much you could add to the rod without moving the reel seat will depend on how long the existing handle is. You said it was short. With a "normal" pre formed butt cork being almost 14 inches long I doubt you would be able to add more than the difference between the existing cork length and the 13.75 inches pre formed corks come in. I will guess at the length of your current cork and say you will only end up adding 4 inches give or take to your rod with a lot of work. The other option is to build your own cork by glueing up pieces of cork and lathing out the shape handle you want, but the even adds more work to the project.
If you do remove your cork, add a short piece on to the rod, build up the lower end of the rod until the taper is removed, you will need to drill a straight hole thru the new cork and slip it on from the bottom.
Removing a reel seat from a rod is quite difficult if it was installed properly. Which means working from the top side of the rod is almost out of the question, but not impossible.
Good luck.