A lot of people expect to negotiate the salary of a new job offer, and there are plenty of other benefits on the table too.

But have you ever thought about negotiating your exit from the company?

Doing so helped Tessa White, a career coach with 20 years of HR experience and founder of The Job Doctor, weather a layoff earlier in her career.

"The smartest thing I ever did was, I pre-negotiated my severance agreement," White tells CNBC Make It.

At the time, White was up for a head of HR position for UnitedHealth Group.

She didn't anticipate they'd actually change their severance policy, "but I was a single parent raising three children at the time, and I wanted to make sure if I [joined the company], that I would be protected from unknowns," she says.

DON'T MISS: The ultimate guide to negotiating a higher salary

She likens it to when a couple gets a prenuptial agreement before marriage. "When they love you at the company" and want to hire you "is the time to negotiate your severance," she says.

Her pre-negotiated severance paid off

By negotiating her severance early on, White says she could assure her new boss she was committed to the job while also future-proofing her career and finances.

She told her hiring manager, "Look, I have a lot of confidence in myself. I know I will do a good job for you," she says. "I'm not asking for severance in the event I'm terminated for performance."

Then, she made the case for additional financial security. She continued: "I'm taking a chance on something that's really steady for me now to go into a new company, and I want to make sure that I am able to have the security that I'll land on my feet if the unexpected happens" like a merger, acquisition or another shift in business priorities.

The company agreed to extend her severance pay from two months of coverage to four in the event White was let go for reasons unrelated to her performance.

As it happens, the company changed ownership several years later, moved its headquarters and asked White to move or lose her job with severance.

Knowing she'd secured extra severance coverage "made all of the panic go away when I heard the big changes that were ahead," she says. "I knew I could care for my family and land on my feet."

She now passes along the advice to others: "I tell my clients to do that all the time," she says, and "every time I've had a client ask for that, they've gotten it."

What you can negotiate in a severance agreement

Certain terms of severance agreements can be negotiated once a layoff is initiated, but there isn't always a ton of flexibility. Companies generally have a policy around how much severance they pay out according to the employee's years of service, for example, but it could be worthwhile to ask for longer coverage if you're a star performer or in a senior role.

Severance is paid either as a lump sum or in installments, and you could negotiate for one method over another depending on your financial situation.

You may have more wiggle room negotiating the number of months or amount of coverage the company will provide through COBRA, White says.

Other post-layoff negotiations are less likely unless your termination is risky for the company and could result in a lawsuit.

Generally, White says, it's much easier to have the severance discussion before a layoff actually happens, and "hard on the back-end, if not impossible."

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