Deep polar codes are pre-transformed polar codes that employ a multi-layered polar kernel transformation strategy to enhance code performance in short blocklength regimes. However, like conventional polar codes, their block length is constrained to powers of two, as the final transformation layer uses a conventional polar kernel matrix. This paper introduces a novel rate-matching technique for deep polar codes using code extension, particularly effective when the desired code length slightly exceeds a power of two. The key idea is to exploit the layered structure of deep polar codes by concatenating polar codewords generated at each transformation layer. Based on this structure, we also develop an efficient decoding algorithm leveraging soft-output successive cancellation list decoding and provide comprehensive error probability analysis supporting our code design algorithms. Additionally, we propose a computationally efficient greedy algorithm for multi-layer configurations. Extensive simulations confirm that our approach delivers substantial coding gains over conventional rate-matching methods, especially in medium to high code-rate regimes.

13 pages, 11 figures