That would likely not warrant the trouble or expense for an increase of 300 MHz on peak turbo boost clocks; of course, if you had a buyer for your old processor, and, found a sale on the old one making it only a ~$20 expense for the entire upgrade...
Of course, the 9600K does not come with a cooler, and, I'd not particularly expect the cooler from the 8600 to be adequate, frankly...
If jumping to a 2080Ti (Ti is darned expensive!), then you'd want at least a 9700K, IMO...; and, with the 9900K only $70-$80 more....might as well go for it.