I measured the dimensions of the loop portion of the OEM jack (my name for the female part of the jack that receives the hook end of the multi-part handle) – those dimensions are shown in the attached photo. Roughly speaking, the width ranges from 0.95 – 0.98”, the thickness ranges from 0.52 – 0.58”, and the hole measures about 0.52 – 0.54” across.
View attachment 2709987
I also measured the dimensions of the hook itself, and of the female end of that segment of the multi-part handle. The hook diameter is roughly 0.48 – 0.50”, and the female opening on the end of that piece of the multi-part handle is approximately 0.44” x 0.44”.
That last data point would imply that a 3/8” square drive would -almost- fit, as it measures about 0.37” x 0.37”. If you could get it to mate securely, you could use a battery operated drill or impact driver (with or without an extension) and a 3/8” square drive tip to turn the hook segment from the OEM tool kit. However, my test with a 3/8” square drive bit showed that it was a bit too loose of a fit, creating the risk that you might spin the square drive in the female end of the factory hook segment, destroying that end and rendering it useless.
Instead, I think an appropriately sized ½” drive “universal socket” might just do the trick. This one from McMaster-Carr has 54 pins and has the ability to receive 7/16” – 1¼” (0.4375” – 1.2500”) square, hex, 8-point fastener heads, so I would think that it could receive and grip/turn the 0.5” x 0.9” (external dimensions) loop on the jack.
McMaster-Carr - https://www.mcmaster.com/6021A11/
Add an appropriate length extension and your electric driver of choice and it just might work. I’ll be purchasing one of these as soon as I see one locally, and will report back on whether or not it works.