Here are the two big things.
Promoting healthy lifestyles to lower overall cost.
Creating a sliding scale system of premiums to tax higher income people and subsidize insurance premiums for lower income people. The "income transfer" component.
A healthier population can mean less burden, cost wise for the entire system, be it public or private. Promote healthier lifestyles in social planning. Bike-able neighborhoods, less smoking and better diets for instance. This will make it easier to extend coverage to more people without bankrupting society.
The other thing to remember is to create a sliding scale system so lower income people can still afford the coverage. This most likely means (no matter how we wallpaper it) income transfer.
Yes, the gap between upper income and lower income people has gotten so wide that universal coverage is no longer possible (or at least no longer probable) unless there is some sort of tax on the wealthy to subsidize the sliding scale for lower income. The tax would probably also hit upper middle class, but it's essential for universal coverage to be a reality.
We already have this "income transfer" to some extent. For instance, hospitals "cost shift" by charging folks with insurance, or the ability to pay, more than the true cost of the procedure. This margin helps to make up for the loss caused by providing uncompensated care to those who can't afford to pay, or don't have insurance.
Since cost shifting is already happening, it could be done more rationally. Low income people often are made to get their care from the emergency room which is an expensive part of the health care delivery system.
If we don't have some kind of income transfer / cost shift, low income people are likely to just be turned away from health care. We could head into an environment of euthanasia for lower income people. At least those who have both low income and high medical expense.
Other aspects of reform can wring savings out of the system, I'm sure.
Things like tort reform, but the two main things, in my opinion, are healthier lifestyles and some kind of sliding scale for premiums.