This paper investigates the design of spatially coupled low-density parity-check (SC-LDPC) codes constructed from connected-chain ensembles for bit-interleaved coded modulation (BICM) schemes. For short coupling lengths, connecting multiple SC-LDPC chains can improve decoding performance over single-chains and impose structured unequal error protection (UEP). A joint design of connected-chain ensembles and bit mapping to further exploit the UEP from codes and high-order modulations is proposed. Numerical results demonstrate the superiority of the proposed design over existing connected-chain ensembles and over single-chain ensembles with existing bit mapping design.