Unless you have a contract saying otherwise, companies that lay you off aren't legally required to provide you with severance. Typically, though, they will offer something. The question is whether it's enough given your tenure and contributions.
If you don't think it is, you may be able to negotiate a better package if you have some points of leverage. A key one is what New York-based compensation attorney Alan Sklover calls "pipeline value." That is, are you the only person at the company familiar with important details of a current project or who knows how to perform a critical function for the employer?
In other words, do they still need something from you?
What you should do: When making your request, keep in mind, you're negotiating with individuals, not "the company," Sklover said. And those individuals have things to lose, including their bonuses or jobs, if negotiations with you don't go well.
The best way to make your request: Respectfully, in writing, to someone in a position to give you more severance -- i.e., not HR. You might start with your manager, or your manager's manager.
Sklover also has recommended clients send their letters directly to the board of directors if they have a legitimate claim and those lower down on the totem pole aren't persuaded.