Once you've hit your fiscal capacity to tax the rich, a few big sources of tax revenue are left:
1) A value-added tax. Very efficient and generates a lot of money because evasion is very difficult; it is almost self-enforcing. It minimizes economic distortion, and what distortions it does introduce encourage savings over consumption. It is also highly regressive. So it's hard to see where the political support will come from: Progressives hate the regressivity, conservatives, the imposition of a large new tax that will squeeze a lot of money out of people.
2) Raising income taxes on the middle class. Also raises a lot of money -- the middle class mostly have salary income, and they don't have the ability of the wealthy to shift their income between, say, capital gains and ordinary income. This will raise a lot of money (note that the Bush tax cuts on the middle class, which were made permanent in 2010, cost about three times as much as those on the wealthy.) It will also rile many millions of people, who will make angry phone calls to their representatives.
3) Tax the savings of the middle class. This could take many forms: lowering the dollar value of an estate that is exempted from tax, eliminate the basis-step up that such estates currently enjoy, or start to pare back on tax-advantaged savings like Roth IRAs and 529 educational savings accounts. (Traditional IRAs and 401(k)s already have their withdrawals taxed as ordinary income, which will make it harder for the government to claw back the tax benefit they've already extended without outright seizing the accounts.) This will also cause a mass freakout, but of a smaller number of people, since a surprising number of affluent people save very little of their income.
The third option is the worst, from an economic point of view -- the last thing we want to do is discourage saving, given how little of it Americans do. On the other hand, it may be the most politically palatable.