When is the best time of year to transplant tiger lilies?

My mother was an avid flower collector especially of day lilies of which she had over 300 different named & seedling varieties. Just a side note, but she had a multitude of other flowers in our 10 or so beds, but loved the lilies most. However, after her death in 1997, my dad has had little success or desire maintaining the vast array that Mom created. Before he plows over all of the beds that are now overgrown, all 4 of us kids want to be sure we have a few of our favorites and one of my top priorities is relocating the tiger lilies from my grandmother's farm. I'll be moving them about 150 miles to my home outside of Nashville, Tennessee and don't want to do it at the wrong time or under the wrong conditions to threaten the success of the transplant. Advice is appreciated!