Quantum teleportation is a foundational protocol for sending quantum information through entanglement distribution and classical communication. Assuming ideal classical communication, the reliability of quantum teleportation is limited by the fidelity of the shared EPR pairs. This reliability can be improved through two mechanisms: entanglement purification and quantum error correction (QEC). Using both techniques in concert requires flexible QEC rates, since purification alters the structure of errors induced by imperfect-EPR teleportation, and fixed-rate codes cannot be uniformly effective across purification regimes or reliability targets. In this work, we supplement purification with punctured QEC codes, providing a family of code variants that can be adapted to error-channel characteristics and reliability targets. Punctured codes improve teleportation reliability across a broader range of purification regimes, enabling target reliability to be met without hardware-level code switching. This is corroborated by numerical results, showing that different punctured codes achieve the lowest logical error probability in different operating regimes, and that selecting among them reduces logical error relative to fixed-rate encoded teleportation. This reduction relaxes the requirement on the initial EPR fidelity or purification needed to achieve a target reliability. Overall, puncturing enables adaptation to varying entanglement conditions and reliability requirements while reusing a single stabilizer structure.