Yes, relativity still holds for everyday, slow speeds. Relativity differs appreciably from classical mechanics only for fast speeds (near the speed of light).
Work Step by Step
The correspondence principle requires that old (classical mechanics) and new (relativity) theories agree in the area of overlap, where the old theory has been well-verified. If a new theory is valid, it must give the same correct results as the old theory, i.e., it must correspond, in any area where the old theory has been verified. This is discussed on page 681.