The best time to transplant hostas and other perennials is spring or fall when the weather is cool and there is more natural moisture in the soil. In most areas there's a 4 week window in the spring when the ground thaws and in fall before the first hard frost.
For most perennials, spring is the best time to transplant. In the spring, plants need to be transplanted while dormant before they begin actively growing. In the fall, dead growth needs to be cut back before transplanting to allow the plant to focus on establishing roots before the ground freezes. Transplants also need plenty of water to make sure that roots develop to support the plant before the heat of summer and the freeze of winter.