The only time not giving extra notice is a bad idea is if you suspect your old employer will react badly. Such as:
If you already have vacation planned during the transition period that they would refuse to let you take, or want to take some time off at short notice because you'll forfeit unused leave when you leave. In this case, keep quiet until you've returned from vacation/etc.
If they might immediately show you the door instead of allowing you to work out your notice period in the office; and you can't start early with your new employer (because they can't bring you on sooner, moving logistics mean you can't start early, etc). In this case, whether you wait until the last minute or announce early with the hope of being able to start your new job sooner, you should still put together a document containing any key information your successor will need and making your ex boss of it's location. Just because senior management are jerks is no reason to spite the other people you worked with.
