The last time a went from one system administrator job to another, I proposed a very simple plan...
I told my employer that if they would hire the new sysadm with a month overlap, I would do my best to train him in everything. I would even allow the new admin to call me from time to time over the next few months, if he panicked. I asked my new employee if they we're okay with me answering a few phone calls from time to time, they saw no problem at all in that.
It all played out very well. They hired a new guy with a month of overlap as I proposed, but it didn't take longer than a week or two before he was up to speed. My employer suggested than in return for them to be able to call me in the future, I could take the last time off as paid vacation. Great!
They eventually did call ~6 month later - not because they we're in panic, but they had a difficult problem that needed solving and they thought of me. I talked to my new employer, and we we're able to help them out within a few hours. Everyone was happy! :-)
Moral of the story; don't be a prick. If you treat people nice and give them a fair way out, chances are the won't bother you.