In the versatile video coding (VVC) standard, cross-component prediction (CCP) is introduced to utilize the correlation between luma and chroma components. To further exploit the cross-component redundancy, a bundle of novel CCP methods with more sophisticated models are studied in enhanced compression model (ECM), which is a platform targeting at the next generation video coding standard, developed by JVET. This paper presents two methods, known as local-boosting CCP (LB-CCP) and non-local CCP (NL-CCP) to improve CCP with local and non-local information. With LB-CCP, prediction samples of CCP can be filtered with neighbouring samples. Besides, neighbouring template costs are calculated to determine the range of training samples, as well as the cross-component prediction method used in the chroma fusion mode. With NL-CCP, a CCP model can be derived with samples non-adjacent to the current block. As an equivalent but simpler implementation, the CCP model can be inherited from a CCP-coded neighbouring block as a spatial CCP candidate. Moreover, the CCP model of a previous CCP-coded block can be stored in a history-based table, which can be fetched and used by the current block as a history-based CCP candidate. A NL-CCP candidate list is built with the two kinds of candidates. The encoder can select the optimal candidate and send an index to the decoder. Experimental results show that LB-CCP together with NL-CCP can provide an average Bjontegaard delta rate (BD-rate) reduction of 0.27%, 2.31%, 2.44% on Y, Cb, Cr components, respectively, with a negligible change in the running time, compared with ECM-8.0 in all intra configurations. Both LB-CCP and NL-CCP have been adopted into ECM.