I think that SynergEyes is the brand of contact lenses you were referring to. They are hard lenses in the center with a soft lens "skirt." They are supposed to be more comfortable than conventional GP contacts, and they can correct astigmatism. I don't know whether they would work for you, and I don't know anyone who has tried them. You can read about them online.
There is a new experimental treatment for k conus called cross linking. It has been discussed here and also on the internet. I can't tell if your cornea is too far gone for treatment.
You may need to see a highly specialized contact lens fitter either Eye MD or optom that does toric and bi toric contact fitting for k conus.
JCH MD