You should be able to apply for a job that is longer than 3 months, and once you have the offer get the necessary forms from the Ausländerbehörde for you and your employer to complete for the work permit. Once you have this you can then also apply to have your residence permit extended.NB these are TWO separate permits - the residence permit will be a stamp in your passport and the working permit is a separate piece of paper. One is no good without the other.
Also be aware that due to the unemployment levels in Germany the Behörde may check to see if there is someone in the current unemployment database that can do the job first, before issuing you a permit. It helps if you can get the employer to say that they need a native English speaker for whatever reason. Not sure if this will work in your profession, or if there is an overabundance of chefs, but be aware that it is a potential hurdle.
Basic info available always from the German Consulate in Australia!
German Consulate Melbourne - work visa
It can take a while sometimes to get things processed, if you don't use a relocation firm so do not wait until the end of your current permit to apply.
Ausländerbehörde
EDIT: Just noticed from your profile that you are in Hamburg. Just search google for Ausländerbehörde Hamburg and you should be on the right path.
Try this for Hamburg (seems to only be in German so let us know if you need help with it)
Ausländerbehörde Hamburg
Good luck!!!