In this paper, we present a new class of spatially coupled codes obtained by using both non-recursive and recursive block-oriented superposition. The resulting codes are termed as bidirectional block Markov superposition transmission (BiBMST) codes. Firstly, we perform an iterative decoding threshold analysis according to protograph-based extrinsic information transfer (PEXIT) charts for the BiBMST codes over the binary erasure channels (BECs). Secondly, we derive the generator and parity-check matrices of the BiBMST codes. Thirdly, extensive numerical results are presented to show the advantages of the proposed BiBMST codes. Particularly, our numerical results show that, under the constraint of an equal decoding latency, the BiBMST codes perform better than the recursive BMST (rBMST) codes. However, the simulation results show that, in finite-length regime, negligible performance gain is obtained by increasing the encoding memory. We solve this limitation by introducing partial superposition, and the resulting codes are termed as partially-connected BiBMST (PC-BiBMST) code. Analytical results have confirmed the advantages of the PC-BiBMST codes over the original BiBMST codes. We also present extensive simulation results to show the performance advantages of the PC-BiBMST codes over the spatially coupled low-density parity-check (SC-LDPC) codes, spatially coupled generalized LDPC (SC-GLDPC) codes, and the original BiBMST codes in the finite-length regime.