In the case of IE6, it would mean setting up some machines with $NEXT_GEN_BROWSER and some side servers with copies of the antiquated "web" apps. Gradually, everything that's broken without IE6 is fixed by tinkering with the side copy that no one is really using. Sometimes, the company's been really stupid and doesn't even have source code, etc. so "tinkering" might really mean "try something new entirely", but at least they'd be showing interest in modernizing (however long it takes).
The simple fact is that all companies should have this kind of "beta flow", in which they can basically try whatever they want (time permitting). Today it's IE6, tomorrow it could be something entirely different; there will always something new that "might" vastly improve efficiency, and a company that can give these things an honest trial is way ahead. I cringe when I see change-averse IT groups, because deep down I know they're shooting themselves in the foot.