Possible explanations for a hissing toilet
There should be a tube coming from the ballcock valve (Korky unit) going to the overflow pipe with a clip holding it onto the edge of the pipe. The purpose of that is to run some water down directly to the bowl after the flapper closes in order to refill the bowl (and help take care of "Klingons" on the side). I have found Korky fill valves to be especially susceptible to leaking through to that bowl refill tube continuously. The water runs along the side of the overflow pipe, so it's hard to see. You don't see anything going over the top of the overflow pipe and if you put dye in the tank you don't see it leaking. But if you pull that bowl refill tube off of the clip, you will see a small stream or drip of water coming out.