Unless you have some magical political pull, your leverage is nil. The reality is, getting to the point that the fortune 50 or .gov is suing you means that you were asleep at the switch -- getting to that point means that the business unit you are working with has acknowledged defeat (a huge step) and is going to blame you and hurt you no matter what the contract says.
also it's way easier to negotiate the "definition of done" with them. typically if all of their requirements are addressed and bugs reported in UAT are fixed by the go-live they are good. their teams take over and unless you're hosting the system (which would be covered by a separate hosting agreement) - you're pretty much done.
also smaller companies tend to demand "warranty" after the go-live (mostly due to the fact they don't have the staff to maintain the system).
My first programming job had exactly this problem. Spec started at "connect these two services together using webhooks for $700" and creeped its way up to "build a database-backed web service that replaces both of the original services and then some for $4000, plus development expenses."
It's still not done. It'll never be done. But thankfully, I got out after it was off the ground and someone else maintains it now.