A GPS-disciplined oscillator (GPSDO) provides a cost-effective alternative to a primary clock such as a cesium-beam atomic clock. A rubidium (Rb) atomic oscillator is usually employed in a GPSDO as a holdover clock where maintaining the most accurate time is required during an extended absence of GPS signals. A commercially available Chip-Scale Atomic Clock (CSAC) offers a comparable long-term stability and is an attractive candidate for a holdover clock in certain applications especially where reduced Size, Weight, and Power (SWAP) is desirable. In this paper, the retrace effects on the holdover clock errors of CSAC from a cold start are characterized. The disciplining behaviors with various durations of the disciplining time are also characterized with short and long disciplining time constants in order to seek an optimal time constant.