Typically no, because employees 'sell' their rights to the patent to their employer when filing the patent application. It is customary for the employees to get a silver dollar as a token that has monetary value (hence the sale) and we can thank Richard Feynman for forcing there to be an actual transfer of money :)

I should also add that usually the inventor can't take the patent to their next job but if the inventor wrote super-narrow claims into the patent then that would be easy to work around at a future employer.

I don't think this is necessary. You can sign the rights away without a transfer of money.

You always may rightfully say you're the inventor (or one of several) of a patent issued for something you invented.

But you can't "use it" unless you own rights to the invention. Patent inventorship and ownership are different things.

Most likely, you assigned all rights to your invention to your employer at the time of invention. The patent will name you as the inventor, and your then-employer will have all rights to the invention.

Neither you nor your next employer can use the patent unless you negotiate permission from the rights holder (your former employer or whomever they sold the rights to).