I think you might mean "yet." You are starting there, but you haven't started there yet. I don't yet work for that company.
Using "still" is a completely different meaning. You used to work there, but you don't work there anymore. I don't still work for that company.
You can use already, but it's not very common, at least in my experience. You can use it like this: Don't tell me you have eaten already! I planned to take you out to dinner. (Funny how both Clive and I thought of food examples.)
In the company example, that's present, not present perfect. Already is common with negative present. If you don't already have your books open to page 3, please do that now. If you don't already have your seatbelt on, please fasten it so we can get going.