We consider reduced generalized jet schemes LZ(Y) along Z of a infinitesimal deformation Y of a complete intersection variety X over the same fat point Z. By definition, these reduced generalized jet schemes LZ(Y) live over the so-called reduced auto-arc spaces AZ, which were originally introduced by H. Schoutens and later studied by the author. Some well-known results of M. Mustaţǎ on classical jet schemes of locally complete intersections carry over to this case. For example, as a consequence of miracle flatness and some generalizations of the aforementioned results of M. Mustaţǎ, it is shown that the reduced generalized jet space, denoted Ln(Xn), of a flat deformation Xn→Dn along ▪ is usually a global flat deformation over Akn of the classical jet scheme Ln(X) provided the base scheme X is a locally complete intersection and the fibers are all of minimal dimension. Similar interesting relativized versions of those aforementioned results are also explored. Also, the initiation of the so-called regulated defect of a formal deformation in analogy to log canonical threshold is introduced. In the end, the situation is studied when the base scheme X is an algebraic curve, and the notions of so-called strong/weak deformations is introduced in this context. Finally, a so-called deformed motivic volume is defined.