As Bunnett explored improvisation, her interests moved toward the more progressive sounds of the times. She was strongly influenced by the music of Eric Dolphy and Steve Lacy, the master of the soprano saxophone; eventually, supported by a Canadian government grant, she was able to study with Lacy in Paris. Another grant took her to New York, where she apprenticed with Don Pullen, a pianist and composer who combined the spirit of the avant-garde with a passion for the whole jazz tradition. She collaborated with Pullen on her first recording, 1989's New York Duets. She followed it the next year with a quintet release that also featured her husband and bandmate, trumpeter Larry Cramer.
But by this time she was already moving in a very different direction. In 1982 she and her husband took a vacation in Cuba, and the experience literally changed their lives. Bunnett discovered a rich musical world largely untouched by homogenizing commercial pressures. The couple kept returning to Cuba, discovering new musicians, helping deliver instruments to the impoverished country, and learning the complex rhythms and melodic patterns of the island. Nine years after their first visit, they released Spirits of Havana, with local musicians. This was not "world music" muzak, but a meeting of artistic traditions, and it pointed the way for a new direction in Bunnett's musical search. The CD title became the name of her new band, which includes Cramer and Cuban players resident in Toronto.