Taylor Mountain is the mountain you can see to the South East from Idaho Falls. Most of the "off-road" area is private and posted with "No Trespassing" signs. However, there are some trails and the Taylor Creek Road that is open to use for running and hiking. To get to the Taylor Creek Road, you go South on Ammon/Lincoln road. After about 4-5 miles of dirt road there is an intersection, if you go straight you go to an abandoned ski resort (which is posted with "No Trespassing" signs) on "Taylor Mountain Road", but if you turn to the South on the intersection the road is "Taylor Creek Road" and is open to vehicle travel. Taylor Creek Road eventually connects to Wolverine Canyon Road.
Taylor Creek Road itself, especially when it is raining, is a great place to train just a short distance from Idaho Falls.I've never seen another person up there before 8:00 AM and even in the afternoon, it's rare that anyone goes there. There are a few areas where you can leave the dirt road to get some actual "trails" but you have to know where to look.
The road itself isn't maintained for much of the year, so you do have to be cautious about getting stuck in the mud and/or snow and if it's wet at all or there is snow on the ground you will definitely want 4 wheel drive if you plan on driving up the road at all. However, before you get into the Canyon area, the road is suitable for pretty much any car, and then you can run/hike the rest of the way.