My noid vinicolor paphiopedilum recently finished blooming. I've cut off the dried flower spike. Almost immediately after the flower wilted, the orchid started working on a new growth.
Once the growth becomes large enough to be putting out roots of its own, I will attempt to repot the orchid such that the growth is a little close to the potting media. Right now it's quite a few inches up the stem of the previous growth.