Proxy re-encryption (PRE) is a cryptographic primitive that extends public key encryption by allowing ciphertexts to be re-encrypted from one user to another without revealing information about the underlying plaintext. This makes it an essential privacy-enhancing technology, as only the intended recipient is able to decrypt sensitive personal information. Previous PRE schemes were commonly based on symmetric bilinear pairings. However, these have been found to be slower and less secure than the more modern asymmetric pairings. To address this, we propose two new PRE scheme variants, based on the unidirectional symmetric pairing-based scheme by Weng et al. and adapted to utilize asymmetric pairings. We employ a known automated black-box reduction technique to transform the base scheme to the asymmetric setting, identify its shortcomings, and subsequently present an alternative manual transformation that fixes these flaws. The adapted schemes retain the properties of the base scheme and are therefore CCA-secure in the adaptive corruption model without the use of random oracles, while being faster, practical, and more secure overall than the base scheme.