Since multi-edge type low-density parity-check (MET-LDPC) codes were first proposed, the design of MET-LDPC codes has been extensively studied for various applications. However, the existing design rules assume that check node degrees are in the so-called concentrated form which enables one to conveniently find pairs of edge and node distributions satisfying the socket count equalities (SCEs). However, it in return makes the code design conducted in a limited search space, which may lead to a sub-optimal design. This paper proposes a novel design rule for MET-LDPC codes together with a scheme to efficiently sort out invalid distributions, i.e., ones not satisfying the SCEs, in the design process. The proposed design rule does not require check node degrees to be in the concentrated form and can find out optimized MET-LDPC codes without any restriction on the search space. Based on the proposed design rule, MET-LDPC codes are designed across a wide range of code rates on binary-erasure channel and binary-input additive-white Gaussian channel. The designed codes are compared with ones reported in the open literature, which confirms that MET-LDPC codes with the proposed design rule have better threshold performances regardless of channel and code rate even with sets of limited code parameters.