A multi-order broadband mode converter in a ring-core fiber (RCF) using a multi-pitch chirped long-period fiber grating (LPFG), where multiple pitches were introduced in each chirp to further increase the bandwidth, is proposed and demonstrated. The grating parameters were optimized both theoretically and experimentally to achieve broadband mode generation of OAM ± 2 and OAM ± 3 modes by increasing the number of chirps and pitches. The mode conversion efficiency is higher than 90% with a broadband of 57 nm from 1456 nm to 1513 nm and 51 nm from 1573 nm to 1624 nm, corresponding to the second-order OAM mode and third-order OAM mode, respectively. Additionally, the insertion loss is less than 0.8 dB, and the purity is over 90%. The demonstrated mode converter has successfully achieved simultaneous generation of multi-order broadband OAM modes in a RCF for the first time, which has promising potential for application in OAM mode-division multiplexing systems.