Significantly reducing the number of criminals is fairly easy without overhauling the system. Decriminalize drug use and possession. That'll cut the number of trials (or pleas) substantially all by itself, and even more so if you assume that knock-on effects will reduce crime overall, as many drug legalization proponents think it would. Other nonviolent offenses could be cut back as well. Stop putting people in prison because one of them paid the other one for sex, for example. Here in Virginia, you can potentially go to jail for a year (and, obviously, have a trial if you want it) for driving at 80MPH in a zone marked for 70MPH!

Increasing resources for holding trials is also not terribly difficult and doesn't require any sort of overhaul. It's just a problem of money, and not a big one. Just for a random example, it looks like the courts account for about 0.3% of my local county budget and about 1% of my local state budget. We could literally increase court resources by a factor of 10 with only a modest increase in taxes to fund it.