I don't know of any policies that would prevent this kind of thing. The killers in these mass killings are willing to commit the crime knowing full well they'll be caught and spend the rest of their lives in prison [or get the death penalty]. Or they are willing to kill themselves - as in the Connecticut case - or they allow themselves to be shot by police.
This is a person who is willing to die in order to kill large numbers of people. Almost everyone who commits a mass shooting planned it for hours or sometimes weeks - or in the case of Columbine, a year. They're going to be able and willing to do more to evade legal controls.
If you can block a less-motivated person who is only going to be motivated to get a gun for a short period of time, that's going to be an easier task than preventing someone who is motivated to kill large numbers of people.
We can improve the background checks in two ways. Currently the federal background check only applies to gun dealers. If you buy a gun from a private source there is no federal requirement for a federal background check.
In most of the US if you buy a gun from a friend or relative you're not subject to a background check.
Currently, the law says you can't sell a gun [to a mentally ill person] and a mentally ill person can't buy a gun. But in practice the background checks don't work that well because the records are radically incomplete.