Daylilies are tough plants. They can be transplanted anytime but spring and fall are the best times. Repeat blooming dayliles such as 'Stella de Oro' require frequent division about every two years to keep them blooming strongly. It is the new rhizomes that produce the new flowers. Deadheading is critical for good rebloom. If the foliage looks poor after the first bloom phase, it is probably a sign that division is needed to rejuvenate the plant.
Keep well watered after transplanting and during dry periods. Mulch with about two inches of mulch and keep away from the base of the stems.
Here is some information on transplanting daylilies https://www.extension.umn.edu/garden/yard-garden/flowers/growing-daylilies/#planting