Agree with Teryn. There may be an area or two that is more than three miles from where a trail used to be before it was washed out by the 2011 flash floods, but otherwise, it's near impossible to get five miles from any road, let alone a trail. There is a pretty active hiking community in Los Alamos, and consequently there are trails going almost everywhere.
But the first thing I would do is make certain you have a GPS chip that clearly marks the boundaries for 6B, Bandelier, Santa Clara, San Ildefonso, Cochiti, Pajarito, and Los Alamos, so that you don't accidently cross one of those lines in pursuit of elk. That being said, you'll find elk pretty much on the Forest side of every one of those lines. The elk tend to move a lot (much of it back and forth from 6B), so it's all about timing.