Mulligan certainly had an ear for trad/dixieland; you can hear this in the otherwise ultra-cool "Cherry" by the classic Mulligan Quartet with Chet Baker. It's interesting to ponder whether his fondness for counterpoint owed something to the multi-lined approach of dixieland, though counterpoint was, of course, a pretty standard feature of the West Coast school. Mulligan also came to be accepted as a stalwart of the 1950s mainstream giants of jazz. In the famous CBS "Sound of Jazz" television broadcast, he lines up with Coleman Hawkins, Lester Young and Ben Webster to back Billie Holliday. Norman Granz's pairings of him with Ben Webster and Johnny Hodges are likely to appeal to you:
The aspect of Mulligan you refer to came as quite a revelation to me. I knew him first in the 50s from the Birth of the Cool sessions and the Quartet with Chet Baker and saw him as the epitome of cool. I would never have imagined him playing with Hodges or Webster! But we came to find that Mulligan had depths we hadn't realised!