Typically the inventor has to prepare the patent, and it is first to file now. It is then up to the person doing the filing to assign the patent to a company of their choice.

I'd consider filing your own provisional patent ($70), which will never see the light of day unless you continue the application in a year, but will give you the priority date.

Filing, even for a provisional, isn't trivial. To survive the decades-long process, you'll need claims and drawings from the get-go. You'll also need to scour google.com/patents to make sure it hasn't already been invented.

It won't become known for a year-plus after filing that you have this patent application, and by then hopefully the old company will have forgotten about your little technology.

But rather than being sneaky, instead of signing, consider proposing a separation agreement that basically says you're done, everybody's square and nobody's suing anybody for any reason. If they say whoa there, just say you'd prefer not to patent anything because you value your privacy. You're probably hard to replace and I'd never trade a good employee for a patent.

This gives them the choice of firing you (meaning you can file yourself with no contract ramifications) or giving you a free pass on your patent idea.