The sectional kit comes with another metal warp rod that attaches to the new beam with short little texsolv cords. You could attach your existing rod to that one with additional lengths of texsolv. Be sure to give yourself enough length on the new cords, and that the metal rod falls in-between the metal hoops on the sectional beam when it's wound on. At that point you could wind on a warp in the traditional way (back to front or front to back) and not use any additional warp packing material as the threads will wind on in between the metal hoops
I have an AVL Warping Wheel . This measures and tensions each individual 1" section which I then wind onto the sectional beam a section at a time. So easy to warp by myself, regardless of the length/width of the warp....so....I don't use the method I've outlined above. I do have 2 other looms. A Jane table loom and a Woolhouse Tools Carolyn table loom and warp back to front on those similarly to how JST shows.