You can use traditional chords (for example, overlapping third), to which are systematically added one or more extraneous notes. This technique, used by Stravinsky, produces an interesting result: on one side the ear “become aware” of the tonal chord, recognizing the music as not totally alien; on the other, extraneous notes enrich the harmony more than the traditional one. Compared to the
twelve-tone music 
this lacks the (at least ideological) clear break with the past that has so distanced the public from the composers (or vice versa, depending on your point of view).
“Fantasy 1984” 
uses this technique.