The problem is that my new employer does not provide any medical insurance.

Not if I change jobs and my new employer uses a different insurance company.

Will he find a new employer as a free agent?

His new employers would probably even give him another raise.

Her new employer, she quickly learned, was an intelligence officer.

And another plan, even if available through a new employer, might not cover some claims.

Presumably, most of those people would keep working for their new employers.

She'd had no idea that her new employer was waiting!

Can he use those experiences to help his new employer?

When they leave you give them a completed P45 for their new employer.