This paper presents the analysis of Chandra X-ray snapshot observations of a subsample of the extragalactic sources listed in the revised Third Cambridge radio catalog (3CR), previously lacking X-ray observations and thus observed during Chandra Cycle 15. This data set extends the current Chandra coverage of the 3CR extragalactic catalog up to redshift z = 1.0. Our sample includes 22 sources consisting of 1 compact steep spectrum source, 3 quasars (QSOs), and 18 FR II radio galaxies. As in our previous analyses, here we report the X-ray detections of radio cores and extended structures (i.e., knots, hotspots, and lobes) for all sources in the selected sample. We measured their X-ray intensities in three energy ranges, soft (0.5-1 keV), medium (1-2 keV), and hard (2-7 keV), and we also performed standard X-ray spectral analysis for brighter nuclei. All radio nuclei in our sample have an X-ray counterpart. We also discovered X-ray emission associated with the eastern knot of 3CR 154, with radio hotspots in 3CR 41, 3CR 54, and 3CR 225B, and with the southern lobe of 3CR 107. Extended X-ray radiation around the nuclei 3CR 293.1 and 3CR 323 on a scale of few tens of kiloparsecs was also found. X-ray extended emission, potentially arising from the hot gas in the intergalactic medium and/or due to the high-energy counterpart of lobes, is detected for 3CR 93, 3CR 154, 3CR 292, and 3CR 323 over a scale of a few hundred kiloparsecs. Finally, this work also presents an update on the state-of-the-art of Chandra and XMM-Newton observations for the entire 3CR sample.