Spring, right after it starts growing again, is the best time. That gives it the longest time to establish itself before winter.

Cutting a 1' square segment out of the middle is not the best idea. Miscanthus tends to 'donut,' that is die out in the middle. The youngest part of the crown is at the outer perimeter of the crown, and you want to use that, rather than the center. When I move established grasses, I divide them. I usually cut out the entire crown, divide it into as many pie-shaped segments as I want, and chop off the pointed inner part..about 1/3 of the length of the segment. This way I'm using the youngest and heartiest part of the crown for the new planting.

Spring? Good to know. I would have guessed Fall. I figured I'd just slice it into four quadrants and replant one of them.
I know about the "donutting". I saw a garden show where someone dug out the middle of the clump, going down into the soil, and replaced the soil in the middle. The idea was that the clump would refill into the center. I think I'll just let it grow.

"I saw a garden show where someone dug out the middle of the clump, going down into the soil, and replaced the soil in the middle. The idea was that the clump would refill into the center."

Very interesting idea, and it should work. I'll have to try that.

I have successfully divided grasses in summer and fall here in Portland, including Miscanthus. I have also successfully moved them. It rains so much in spring that I just don't get these chores done then. Our fall is so long and soil so warm it is different here than other areas.

I have dug out the dead middle of a Miscanthus and replanted it with chunks dug out from the edges. I didn't wait for it to refill itself.

I have extra Miscanthus sinensis var. purpurascens that I just divided and also a few others. My few gardening friends are all grassed out. I hate to compost perfectly good grasses and have not had such good luck mailing them to friends in other states.