In this paper, we consider convertible codes with the locally repairable property. We present an improved lower bound on access cost associated with (r, δ ⩾ 2)-locality, which extends the known lower bound only related tor.We then provide a general construction of convertible codes with optimal access cost which shows that these codes can have super-linear length or maximally recoverable property. Furthermore, we propose explicit constructions of convertible codes with the final code achieving super-linear length or maximally recoverable property when the initial codes have super-linear length or maximally recoverable property respectively. Specifically, compared with known constructions, our construction is applicable to the case δ > 2 for the first time.