Ptychographic X-ray computed tomography is a coherent diffractive imaging method that offers nanometer-scale resolution with quantitative contrast. It offers the possibility to study relatively thick samples by using high energy X-ray photons and exploiting the phase contrast. However, the limited depth of field forces a compromise between resolution and sample thickness. Multi-slice techniques have been used to account for propagation effects within the sample, enabling imaging beyond the depth-of-field limit. Here we introduce and experimentally demonstrate our multi-slice algorithms that allow for the reconstruction of multiple object slices and the incident illumination, as well as the retrieval of unknown object thickness. Additionally, through numerical studies, we show that smaller scanning steps surprisingly increase the depth of field, which can be further extended by the use of multi-slice methods under conditions stated by theoretical expressions. The results presented here will be instrumental for the routine implementation of the technique for X-ray nanotomography.