Absorbing sets are combinatorially defined objects existing in the Tanner graph of a low-density parity-check (LDPC) code that have been shown to cause failures in the iterative message-passing decoder when transmission occurs over the additive white Gaussian noise channel. In this paper, we propose an edge spreading approach to construct high rate array-based spatially-coupled LDPC codes by jointly optimizing the absorbing set spectrum and the minimum distance. By considering general edge spreadings and by considering a larger memory, we show that strictly better codes can be constructed, both in terms of achievable minimum distance for small-to-moderate values of block length and in terms of the number of small absorbing sets.