Congress can call state ratifying conventions to amend the US constitution. We can increase the federal minimum wage and tie it to the rising cost of living and have healthcare as a basic human right. We can also ban corporate political free speech and ban gerrymandering.
You’d need the political will to do it. You’d first have to replace most of congress and most state legislatures, not to mention most of the governors and the President. Constitutional amendments require massive and strong support, and you’re talking about amendments that would eliminate the type of corruption that currently controls the government.
Running all of those campaigns against the oligarchy would be extremely expensive, because the oligarchy wants it that way, and they will outspend you at every election. You won’t find a single republican willing to support the concept, and most democrats won’t, either.
It’s a good idea to make that a goal someday, but as we’ve seen, a constitution is only as good as the justice system and executives enforcing it. The current state of human rights in America is abysmal, and that’s with the Bill of Rights.
In theory you could skip all the federal positions if you had enough support. It’d just be extremely hard. You would need 34 states to call for a convention, in which the federal government couldn’t say no. And any amendments ratified by 3/4 of the states would go into the constitution. Which is a lot of states, but note it also means you don’t have to have 2/3rds of the states support. You could have 51% support in 38 states and it would go through. Getting billionaires to not fund ad campaigns to fight against such happening would be almost impossible though. You’d have to have amendments that most people supported at opportune times. Like convincing people to kill citizens United because Democrats billionaires and Republican billionaires are clearly using their money to get giant sway and making individual voices worth less. Then once there, maybe get other stuff pushed through to limit lobbying. You’d have to get states pissed at the feds really.