Card-based cryptographic protocols provide secure multi-party computations using a deck of physical cards. The most important primitive of those protocols is the shuffling operation, and most of the existing protocols rely on uniform cyclic shuffles (such as the random cut and random bisection cut) in which each possible outcome is equally likely and all possible outcomes constitute a cyclic subgroup. However, a couple of protocols with non-uniform and/or non-cyclic shuffles were proposed by Koch, Walzer, and Härtel at Asiacrypt 2015. Compared to the previous protocols, their protocols require fewer cards to securely produce a hidden AND value, although to implement of such unconventional shuffles appearing in their protocols remains an open problem. This paper introduces “pile-shifting scramble,” which can be a secure implementation of those shuffles. To implement such unconventional shuffles, we utilize physical cases that can store piles of cards, such as boxes and envelopes. Therefore, humans are able to perform the shuffles using these everyday objects. Furthermore, we show that a certain class of non-uniform and/or non-cyclic shuffles having two possible outcomes can be implemented by the pile-shifting scramble. This also implies that we can improve upon the known COPY protocol using three card cases so that the number of cases required can be reduced to two.